circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · preliminaries central angles the...
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![Page 1: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/1.jpg)
Circles
![Page 2: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/2.jpg)
Preliminaries
•A • B•P
•C
_AB is a semicircle_AC is a minor arc_
BAC is a major arc
![Page 3: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/3.jpg)
Preliminaries
•A • B•P
•C
_AB is a
semicircle_AC is a minor arc_
BAC is a major arc
![Page 4: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/4.jpg)
Preliminaries
•A • B•P
•C
_AB is a semicircle_AC is a
minor arc_
BAC is a major arc
![Page 5: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/5.jpg)
Preliminaries
•A • B•P
•C
_AB is a semicircle_AC is a minor arc_
BAC is a
major arc
![Page 6: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/6.jpg)
Preliminaries
•A • B•P
•C
_AB is a semicircle_AC is a minor arc_
BAC is a major arc
![Page 7: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/7.jpg)
Preliminaries
•A
•B
•P
m◦
Definition∠APB is called a central angle.
Relationship between m∠APB and m_AB?
![Page 8: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/8.jpg)
Preliminaries
•A
•B
•P
m◦
Definition∠APB is called a central angle.
Relationship between m∠APB and m_AB?
![Page 9: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/9.jpg)
Preliminaries
Central AnglesThe measure of a minor arc is the same as the central angle cutting offthat arc.
Central Angles
The measure of a major arc is 360◦ less the measure of the centralangle cutting off the associated minor arc.
DefinitionThe minor arc is also called the intercepted arc.
![Page 10: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/10.jpg)
Preliminaries
Central AnglesThe measure of a minor arc is the same as the central angle cutting offthat arc.
Central Angles
The measure of a major arc is 360◦ less the measure of the centralangle cutting off the associated minor arc.
DefinitionThe minor arc is also called the intercepted arc.
![Page 11: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/11.jpg)
Preliminaries
Central AnglesThe measure of a minor arc is the same as the central angle cutting offthat arc.
Central Angles
The measure of a major arc is 360◦ less the measure of the centralangle cutting off the associated minor arc.
DefinitionThe minor arc is also called the intercepted arc.
![Page 12: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/12.jpg)
Arc Addition Postulate
•A
•B
•P
•C
Arc Addition Postulate
If C is on_AB, then
m_AB = m
_AC + m
_CB
![Page 13: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/13.jpg)
Arc Addition
•A •B
•P
•C
•D
Postulate
If C and D are on_AB and m
_AD = m
_BC, then
m_AC = m
_BD
![Page 14: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/14.jpg)
Concentric Circles
•A •B
•P
•X
•Y
120◦
m_AB? 120◦
m_XY? 120◦
Are the arcs congruent?
What we can sayCongruent central angles have the same intercepted arcs andcongruent arcs have the same measured central angles.
![Page 15: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/15.jpg)
Concentric Circles
•A •B
•P
•X
•Y
120◦
m_AB? 120◦
m_XY? 120◦
Are the arcs congruent?
What we can sayCongruent central angles have the same intercepted arcs andcongruent arcs have the same measured central angles.
![Page 16: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/16.jpg)
Concentric Circles
•A •B
•P
•X
•Y
120◦
m_AB?
120◦
m_XY? 120◦
Are the arcs congruent?
What we can sayCongruent central angles have the same intercepted arcs andcongruent arcs have the same measured central angles.
![Page 17: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/17.jpg)
Concentric Circles
•A •B
•P
•X
•Y
120◦
m_AB? 120◦
m_XY? 120◦
Are the arcs congruent?
What we can sayCongruent central angles have the same intercepted arcs andcongruent arcs have the same measured central angles.
![Page 18: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/18.jpg)
Concentric Circles
•A •B
•P
•X
•Y
120◦
m_AB? 120◦
m_XY?
120◦
Are the arcs congruent?
What we can sayCongruent central angles have the same intercepted arcs andcongruent arcs have the same measured central angles.
![Page 19: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/19.jpg)
Concentric Circles
•A •B
•P
•X
•Y
120◦
m_AB? 120◦
m_XY? 120◦
Are the arcs congruent?
What we can sayCongruent central angles have the same intercepted arcs andcongruent arcs have the same measured central angles.
![Page 20: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/20.jpg)
Concentric Circles
•A •B
•P
•X
•Y
120◦
m_AB? 120◦
m_XY? 120◦
Are the arcs congruent?
What we can sayCongruent central angles have the same intercepted arcs andcongruent arcs have the same measured central angles.
![Page 21: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/21.jpg)
Concentric Circles
•A •B
•P
•X
•Y
120◦
m_AB? 120◦
m_XY? 120◦
Are the arcs congruent?
What we can sayCongruent central angles have the same intercepted arcs andcongruent arcs have the same measured central angles.
![Page 22: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/22.jpg)
Midpoints
•A •B
•P
•C
DefinitionThe midpoint of an arc divides the curve into two congruent curves.
![Page 23: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/23.jpg)
Midpoints
•A •B
•P
•C
DefinitionThe midpoint of an arc divides the curve into two congruent curves.
![Page 24: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/24.jpg)
Midpoints
Example
B is the midpoint of_AC. Prove ∆APB ∼= ∆CPB.
•A •C
•P
•B
∠APB ∼= ∠CPB
BP ∼= BP
AP ∼= CP
SAS
![Page 25: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/25.jpg)
Midpoints
Example
B is the midpoint of_AC. Prove ∆APB ∼= ∆CPB.
•A •C
•P
•B
∠APB ∼= ∠CPB
BP ∼= BP
AP ∼= CP
SAS
![Page 26: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/26.jpg)
Midpoints
Example
B is the midpoint of_AC. Prove ∆APB ∼= ∆CPB.
•A •C
•P
•B
∠APB ∼= ∠CPB
BP ∼= BP
AP ∼= CP
SAS
![Page 27: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/27.jpg)
Midpoints
Example
B is the midpoint of_AC. Prove ∆APB ∼= ∆CPB.
•A •C
•P
•B
∠APB ∼= ∠CPB
BP ∼= BP
AP ∼= CP
SAS
![Page 28: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/28.jpg)
Midpoints
Example
B is the midpoint of_AC. Prove ∆APB ∼= ∆CPB.
•A •C
•P
•B
∠APB ∼= ∠CPB
BP ∼= BP
AP ∼= CP
SAS
![Page 29: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/29.jpg)
Midpoints
Example
B is the midpoint of_AC. Prove ∆APB ∼= ∆CPB.
•A •C
•P
•B
∠APB ∼= ∠CPB
BP ∼= BP
AP ∼= CP
SAS
![Page 30: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/30.jpg)
Midpoints
Example
B is the midpoint of_AC. Prove ∆APB ∼= ∆CPB.
•A •C
•P
•B
∠APB ∼= ∠CPB
BP ∼= BP
AP ∼= CP
SAS
![Page 31: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/31.jpg)
Related Statement
•A •C
•P
•B
TheoremIn the same or congruent circles, congruent arcs have congruentchords.
![Page 32: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/32.jpg)
Related Statement
•A •C
•P
•B
TheoremIn the same or congruent circles, congruent arcs have congruentchords.
![Page 33: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/33.jpg)
An Example
Example
Given that CA ∼= DB, prove ∠CAD ∼= ∠DBC.
•P
•A •B
•C •D
![Page 34: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/34.jpg)
Diameter and Chords
TheoremIn a circle, a diameter drawn perpendicular to a chord bisects thechord and its arc.
•P
•A •B
![Page 35: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/35.jpg)
Diameter and Chords
TheoremIn a circle, a diameter drawn perpendicular to a chord bisects thechord and its arc.
•P
•A •B
![Page 36: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/36.jpg)
An Example
ExampleThe length of the diameter of a circle is 20 and the length of the chordAB is 16. What is the shortest distance between the chord and thecenter of the circle?
•P
•A •B
d
8
10d = 6
![Page 37: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/37.jpg)
An Example
ExampleThe length of the diameter of a circle is 20 and the length of the chordAB is 16. What is the shortest distance between the chord and thecenter of the circle?
•P
•A •B
d
8
10d = 6
![Page 38: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/38.jpg)
An Example
ExampleThe length of the diameter of a circle is 20 and the length of the chordAB is 16. What is the shortest distance between the chord and thecenter of the circle?
•P
•A •B
d
8
10d = 6
![Page 39: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/39.jpg)
An Example
ExampleThe length of the diameter of a circle is 20 and the length of the chordAB is 16. What is the shortest distance between the chord and thecenter of the circle?
•P
•A •B
d
8
10
d = 6
![Page 40: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/40.jpg)
An Example
ExampleThe length of the diameter of a circle is 20 and the length of the chordAB is 16. What is the shortest distance between the chord and thecenter of the circle?
•P
•A •B
d
8
10d = 6
![Page 41: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/41.jpg)
Parallel Lines
Parallel LinesIn a circle, parallel lines cut off equal arcs.
Example
Given AB||CD, prove_AC ∼=
_BD
• P
•A •B•C
•D•
M
Think ‘difference of arcs’ ...
![Page 42: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/42.jpg)
Parallel Lines
Parallel LinesIn a circle, parallel lines cut off equal arcs.
Example
Given AB||CD, prove_AC ∼=
_BD
• P
•A •B•C
•D
•M
Think ‘difference of arcs’ ...
![Page 43: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/43.jpg)
Parallel Lines
Parallel LinesIn a circle, parallel lines cut off equal arcs.
Example
Given AB||CD, prove_AC ∼=
_BD
• P
•A •B•C
•D•
M
Think ‘difference of arcs’ ...
![Page 44: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/44.jpg)
Parallel Lines
Parallel LinesIn a circle, parallel lines cut off equal arcs.
Example
Given AB||CD, prove_AC ∼=
_BD
• P
•A •B•C
•D•
M
Think ‘difference of arcs’ ...
![Page 45: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/45.jpg)
More Parallel Lines
Example
If CD||AB and AB is the diameter of the circle, find the measure of_AC
and_
BD if m_
CD = 40◦.
• P•A • B
•C
•D
Both arcs are 70◦.
![Page 46: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/46.jpg)
More Parallel Lines
Example
If CD||AB and AB is the diameter of the circle, find the measure of_AC
and_
BD if m_
CD = 40◦.
• P•A • B
•C
•D
Both arcs are 70◦.
![Page 47: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/47.jpg)
More Parallel Lines
Example
If CD||AB and AB is the diameter of the circle, find the measure of_AC
and_
BD if m_
CD = 40◦.
• P•A • B
•C
•D
Both arcs are 70◦.
![Page 48: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/48.jpg)
Tangents and Secants
DefinitionTangent lines intersect a circle at exactly one point.
DefinitionSecant lines intersect a circle at exactly two points.
•P•A
•C •D
![Page 49: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/49.jpg)
Tangents and Secants
DefinitionTangent lines intersect a circle at exactly one point.
DefinitionSecant lines intersect a circle at exactly two points.
•P•A
•C •D
![Page 50: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/50.jpg)
Tangents and Secants
DefinitionTangent lines intersect a circle at exactly one point.
DefinitionSecant lines intersect a circle at exactly two points.
•P•A
•C •D
![Page 51: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/51.jpg)
You know you love proofs
TheoremA radius drawn to a point of tangency is perpendicular to the tangent.
• P•A
B •
•X
![Page 52: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/52.jpg)
You know you love proofs
TheoremA radius drawn to a point of tangency is perpendicular to the tangent.
• P•A
B •
•X
![Page 53: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/53.jpg)
You know you love proofs
TheoremA radius drawn to a point of tangency is perpendicular to the tangent.
• P•A
B •
•X
![Page 54: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/54.jpg)
The Proof
Proof.By contradictionSuppose PA 6⊥ ←−AB. Then there must be another segment, say PX, suchthat PX ⊥ ←−AB
Since PX ⊥ ←−AB, it must be the shortest distance to AB,But, X is exterior to the circle, so mPA < mPX, a contradiction.Therefore, PA ⊥ AB.
![Page 55: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/55.jpg)
The Proof
Proof.By contradictionSuppose PA 6⊥ ←−AB. Then there must be another segment, say PX, suchthat PX ⊥ ←−ABSince PX ⊥ ←−AB, it must be the shortest distance to AB,
But, X is exterior to the circle, so mPA < mPX, a contradiction.Therefore, PA ⊥ AB.
![Page 56: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/56.jpg)
The Proof
Proof.By contradictionSuppose PA 6⊥ ←−AB. Then there must be another segment, say PX, suchthat PX ⊥ ←−ABSince PX ⊥ ←−AB, it must be the shortest distance to AB,But, X is exterior to the circle, so mPA < mPX, a contradiction.
Therefore, PA ⊥ AB.
![Page 57: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/57.jpg)
The Proof
Proof.By contradictionSuppose PA 6⊥ ←−AB. Then there must be another segment, say PX, suchthat PX ⊥ ←−ABSince PX ⊥ ←−AB, it must be the shortest distance to AB,But, X is exterior to the circle, so mPA < mPX, a contradiction.Therefore, PA ⊥ AB.
![Page 58: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/58.jpg)
Tangent Segment
DefinitionA tangent segment is a line segment that has a point on the tangentline and the point of tangency as its endpoints.
• P•A
B •
![Page 59: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/59.jpg)
Tangent Segment
DefinitionA tangent segment is a line segment that has a point on the tangentline and the point of tangency as its endpoints.
• P•A
B •
![Page 60: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/60.jpg)
Tangent Segment
ExampleThe length of a tangent segment drawn from point B to a circle P is 12units. If the radius of circle P is 5 units, find the distance from thepoint B to the center of the circle.
• P•A
B •
12
5
d = 13
![Page 61: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/61.jpg)
Tangent Segment
ExampleThe length of a tangent segment drawn from point B to a circle P is 12units. If the radius of circle P is 5 units, find the distance from thepoint B to the center of the circle.
• P•A
B •
12
5
d = 13
![Page 62: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/62.jpg)
Tangent Segment
ExampleThe length of a tangent segment drawn from point B to a circle P is 12units. If the radius of circle P is 5 units, find the distance from thepoint B to the center of the circle.
• P•A
B •
12
5
d = 13
![Page 63: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/63.jpg)
Tangent Segment
ExampleThe length of a tangent segment drawn from point B to a circle P is 12units. If the radius of circle P is 5 units, find the distance from thepoint B to the center of the circle.
• P•A
B •
12
5
d = 13
![Page 64: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/64.jpg)
Inscribed and Circumscribed Polygons
DefinitionA polygon is circumscribed about a circle if each of its sides aretangent to the circle.
DefinitionA polygon is inscribed in a circle if each of its vertices are on thecircle.
ExampleA quadrilateral is inscribed in a circle such that the sides divide thecircle into arcs whose measures have the ratio 1 : 2 : 3 : 4. how manydegrees is each arc?
36◦, 72◦, 108◦, 144◦
![Page 65: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/65.jpg)
Inscribed and Circumscribed Polygons
DefinitionA polygon is circumscribed about a circle if each of its sides aretangent to the circle.
DefinitionA polygon is inscribed in a circle if each of its vertices are on thecircle.
ExampleA quadrilateral is inscribed in a circle such that the sides divide thecircle into arcs whose measures have the ratio 1 : 2 : 3 : 4. how manydegrees is each arc?
36◦, 72◦, 108◦, 144◦
![Page 66: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/66.jpg)
Inscribed and Circumscribed Polygons
DefinitionA polygon is circumscribed about a circle if each of its sides aretangent to the circle.
DefinitionA polygon is inscribed in a circle if each of its vertices are on thecircle.
ExampleA quadrilateral is inscribed in a circle such that the sides divide thecircle into arcs whose measures have the ratio 1 : 2 : 3 : 4. how manydegrees is each arc?
36◦, 72◦, 108◦, 144◦
![Page 67: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/67.jpg)
Inscribed and Circumscribed Polygons
DefinitionA polygon is circumscribed about a circle if each of its sides aretangent to the circle.
DefinitionA polygon is inscribed in a circle if each of its vertices are on thecircle.
ExampleA quadrilateral is inscribed in a circle such that the sides divide thecircle into arcs whose measures have the ratio 1 : 2 : 3 : 4. how manydegrees is each arc?
36◦, 72◦, 108◦, 144◦
![Page 68: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/68.jpg)
Angle Measurement: Vertex on the Circle
DefinitionAn inscribed angle has it’s vertex on the circumference of the circleand it’s sides are secants.
• P
x◦
2x◦
Angle Measure
The measure of an inscribed angle is equal to 12 the measure of its
intercepted arc.
![Page 69: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/69.jpg)
Angle Measurement: Vertex on the Circle
DefinitionAn inscribed angle has it’s vertex on the circumference of the circleand it’s sides are secants.
• P
x◦
2x◦
Angle Measure
The measure of an inscribed angle is equal to 12 the measure of its
intercepted arc.
![Page 70: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/70.jpg)
Angle Measurement: Vertex on the Circle
DefinitionAn inscribed angle has it’s vertex on the circumference of the circleand it’s sides are secants.
• P
x◦
2x◦
Angle Measure
The measure of an inscribed angle is equal to 12 the measure of its
intercepted arc.
![Page 71: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/71.jpg)
Examples
ExampleFind x.
•
x◦
••
•
110◦
70◦
![Page 72: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/72.jpg)
Examples
ExampleFind x.
•
x◦
••
•
110◦70◦
![Page 73: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/73.jpg)
Examples
ExampleA triangle is inscribed in a circle so that its sides divide the circle intoarcs whose measures have the ratio 2 : 3 : 7. Find the measure of thelargest angle of the triangle.
•
•
•
•
105◦
![Page 74: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/74.jpg)
Examples
ExampleA triangle is inscribed in a circle so that its sides divide the circle intoarcs whose measures have the ratio 2 : 3 : 7. Find the measure of thelargest angle of the triangle.
•
•
•
•
105◦
![Page 75: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/75.jpg)
Examples
ExampleA triangle is inscribed in a circle so that its sides divide the circle intoarcs whose measures have the ratio 2 : 3 : 7. Find the measure of thelargest angle of the triangle.
•
•
•
•
105◦
![Page 76: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/76.jpg)
Examples
Example
Quadrilateral ABCD is inscribed in a circle. If m∠A = x◦ andm∠B = y◦, find angles C and D in terms of x and y.
•
•
•
• •
A
B
DC
x◦y◦
m∠C = (180− x)◦, m∠D = (180− y)◦.
TheoremOpposite angles of an inscribed quadrilateral are supplementary.
![Page 77: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/77.jpg)
Examples
Example
Quadrilateral ABCD is inscribed in a circle. If m∠A = x◦ andm∠B = y◦, find angles C and D in terms of x and y.
•
•
•
• •
A
B
DC
x◦y◦
m∠C = (180− x)◦, m∠D = (180− y)◦.
TheoremOpposite angles of an inscribed quadrilateral are supplementary.
![Page 78: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/78.jpg)
Examples
Example
Quadrilateral ABCD is inscribed in a circle. If m∠A = x◦ andm∠B = y◦, find angles C and D in terms of x and y.
•
•
•
• •
A
B
DC
x◦y◦
m∠C = (180− x)◦, m∠D = (180− y)◦.
TheoremOpposite angles of an inscribed quadrilateral are supplementary.
![Page 79: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/79.jpg)
Examples
Example
Quadrilateral ABCD is inscribed in a circle. If m∠A = x◦ andm∠B = y◦, find angles C and D in terms of x and y.
•
•
•
• •
A
B
DC
x◦y◦
m∠C = (180− x)◦, m∠D = (180− y)◦.
TheoremOpposite angles of an inscribed quadrilateral are supplementary.
![Page 80: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/80.jpg)
More Tangency
TheoremThe measure of an angle formed by a tangent and a chord drawn tothe point of tangency is equal to half the measure of the interceptedarc.
• P
•A • Bx◦
2x◦
•C
![Page 81: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/81.jpg)
More Tangency
TheoremThe measure of an angle formed by a tangent and a chord drawn tothe point of tangency is equal to half the measure of the interceptedarc.
• P
•A • Bx◦
2x◦
•C
![Page 82: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/82.jpg)
More Tangency
Example
Given that BC is tangent to circle P at point B, prove the measure of
angle x is half of the measure of_AB.
• P
•A • Bx◦
•C
•D
AD||BC
![Page 83: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/83.jpg)
More Tangency
Example
Given that BC is tangent to circle P at point B, prove the measure of
angle x is half of the measure of_AB.
• P
•A • Bx◦
•C
•D
AD||BC
![Page 84: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/84.jpg)
More Tangency
Example
Given that BC is tangent to circle P at point B, prove the measure of
angle x is half of the measure of_AB.
• P
•A • Bx◦
•C
•D
AD||BC
![Page 85: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/85.jpg)
Examples
Example
Find the value of x if m_
ACB = 250◦
• P
•A • C
•B
x◦
m_
ACB = 250◦
m_AB = 110◦
x =12
(110◦) = 55◦
![Page 86: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/86.jpg)
Examples
Example
Find the value of x if m_
ACB = 250◦
• P
•A • C
•B
x◦
m_
ACB = 250◦
m_AB = 110◦
x =12
(110◦) = 55◦
![Page 87: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/87.jpg)
Examples
Example
Find the value of x if m_
ACB = 250◦
• P
•A • C
•B
x◦
m_
ACB = 250◦
m_AB = 110◦
x =12
(110◦) = 55◦
![Page 88: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/88.jpg)
Examples
Example
Find the value of x if m_
ACB = 250◦
• P
•A • C
•B
x◦
m_
ACB = 250◦
m_AB = 110◦
x =12
(110◦) = 55◦
![Page 89: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/89.jpg)
Examples
Example
Find the value of x if m_
ACB = 250◦
• P
•A • C
•B
x◦
m_
ACB = 250◦
m_AB = 110◦
x =12
(110◦) = 55◦
![Page 90: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/90.jpg)
Example
Example
Find the measure of the numbered angles if m_AC = 110◦.
• P ••
•
A B
C1 2
3 4 5
6
![Page 91: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/91.jpg)
Example
m∠1 = 12(110) = 55◦
m_
BC = 180− 110 = 70◦ ⇒ m∠2 = 12(70) = 35◦
m∠3 = 35◦
m∠4 = 110◦
m∠5 = 55◦
m∠6 = 12(
_AB) = 1
2(180) = 90◦
![Page 92: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/92.jpg)
Chord-Chord-Angle Theorem
TheoremThe measure of an angle formed by two chords intersecting in theinterior of a circle is equal to 1
2 the sum of the intercepted arcs.
• P•
•
A
B•
•C
D
![Page 93: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/93.jpg)
Example
ExampleFind the value of x.
• P•
•
A
B•
•C
Dx
65◦105◦
x = 85◦
![Page 94: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/94.jpg)
Example
ExampleFind the value of x.
• P•
•
A
B•
•C
Dx
65◦105◦
x = 85◦
![Page 95: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/95.jpg)
Example
ExampleFind the value of x.
• P•
•
A
B•
•C
D108◦62◦
x
x = 154◦
![Page 96: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/96.jpg)
Example
ExampleFind the value of x.
• P•
•
A
B•
•C
D108◦62◦
x
x = 154◦
![Page 97: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/97.jpg)
Activity; The Magic Circle
Materials:
1 one 7 inch circle2 pencil3 ruler (optional)
The reason that the ruler is optional is because you can, at any time,include the measurement of area, perimeter, surface area, or volume.
![Page 98: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/98.jpg)
Activity; The Magic Circle
Materials:
1 one 7 inch circle2 pencil3 ruler (optional)
The reason that the ruler is optional is because you can, at any time,include the measurement of area, perimeter, surface area, or volume.
![Page 99: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/99.jpg)
Activity; The Magic Circle
Step 1Look at the shape you are holding. What is it?
circle
What is the distance around the outside called? circumference
![Page 100: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/100.jpg)
Activity; The Magic Circle
Step 1Look at the shape you are holding. What is it? circle
What is the distance around the outside called? circumference
![Page 101: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/101.jpg)
Activity; The Magic Circle
Step 1Look at the shape you are holding. What is it? circle
What is the distance around the outside called?
circumference
![Page 102: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/102.jpg)
Activity; The Magic Circle
Step 1Look at the shape you are holding. What is it? circle
What is the distance around the outside called? circumference
![Page 103: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/103.jpg)
Activity; The Magic Circle
Step 2Fold your circle directly in half and crease it well.
Open the circle. What is this crease? diameter
![Page 104: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/104.jpg)
Activity; The Magic Circle
Step 2Fold your circle directly in half and crease it well.
Open the circle. What is this crease?
diameter
![Page 105: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/105.jpg)
Activity; The Magic Circle
Step 2Fold your circle directly in half and crease it well.
Open the circle. What is this crease? diameter
![Page 106: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/106.jpg)
Activity; The Magic Circle
Step 3Hold the circle at the ends of the crease. Fold your circle in half again,but this time match up the end points of the crease.
Open your circle, is this also a diameter? How do you know?
Is there something special about the way these lines intersect?The lines are perpendicular
![Page 107: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/107.jpg)
Activity; The Magic Circle
Step 3Hold the circle at the ends of the crease. Fold your circle in half again,but this time match up the end points of the crease.
Open your circle, is this also a diameter? How do you know?
Is there something special about the way these lines intersect?The lines are perpendicular
![Page 108: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/108.jpg)
Activity; The Magic Circle
Step 3Hold the circle at the ends of the crease. Fold your circle in half again,but this time match up the end points of the crease.
Open your circle, is this also a diameter? How do you know?
Is there something special about the way these lines intersect?
The lines are perpendicular
![Page 109: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/109.jpg)
Activity; The Magic Circle
Step 3Hold the circle at the ends of the crease. Fold your circle in half again,but this time match up the end points of the crease.
Open your circle, is this also a diameter? How do you know?
Is there something special about the way these lines intersect?The lines are perpendicular
![Page 110: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/110.jpg)
Activity; The Magic Circle
Step 4Place a dot, no bigger than the width of a pencil, at the point wherethe creases connect.
What is this point called? center
![Page 111: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/111.jpg)
Activity; The Magic Circle
Step 4Place a dot, no bigger than the width of a pencil, at the point wherethe creases connect.
What is this point called?
center
![Page 112: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/112.jpg)
Activity; The Magic Circle
Step 4Place a dot, no bigger than the width of a pencil, at the point wherethe creases connect.
What is this point called? center
![Page 113: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/113.jpg)
Activity; The Magic Circle
Step 5Using your pencil, trace one of the lines from the center to the edge ofthe circle.
What is this line called? radius
![Page 114: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/114.jpg)
Activity; The Magic Circle
Step 5Using your pencil, trace one of the lines from the center to the edge ofthe circle.
What is this line called?
radius
![Page 115: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/115.jpg)
Activity; The Magic Circle
Step 5Using your pencil, trace one of the lines from the center to the edge ofthe circle.
What is this line called? radius
![Page 116: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/116.jpg)
Activity; The Magic Circle
Step 6Fold in one of the outer, curved edges of the circle until it just touchesthe dot in the middle. Crease it well.
Open the fold and look at the crease you just made. Is it a diameter?Is it a radius? Why or why not?
What is this segment called? chord
![Page 117: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/117.jpg)
Activity; The Magic Circle
Step 6Fold in one of the outer, curved edges of the circle until it just touchesthe dot in the middle. Crease it well.
Open the fold and look at the crease you just made. Is it a diameter?Is it a radius? Why or why not?
What is this segment called? chord
![Page 118: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/118.jpg)
Activity; The Magic Circle
Step 6Fold in one of the outer, curved edges of the circle until it just touchesthe dot in the middle. Crease it well.
Open the fold and look at the crease you just made. Is it a diameter?Is it a radius? Why or why not?
What is this segment called?
chord
![Page 119: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/119.jpg)
Activity; The Magic Circle
Step 6Fold in one of the outer, curved edges of the circle until it just touchesthe dot in the middle. Crease it well.
Open the fold and look at the crease you just made. Is it a diameter?Is it a radius? Why or why not?
What is this segment called? chord
![Page 120: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/120.jpg)
Activity; The Magic Circle
Step 7Look at the curved part of the circle between the points where thisline touches the outside of the circle.
What is this curved part called? arc
![Page 121: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/121.jpg)
Activity; The Magic Circle
Step 7Look at the curved part of the circle between the points where thisline touches the outside of the circle.
What is this curved part called?
arc
![Page 122: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/122.jpg)
Activity; The Magic Circle
Step 7Look at the curved part of the circle between the points where thisline touches the outside of the circle.
What is this curved part called? arc
![Page 123: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/123.jpg)
Activity; The Magic Circle
Step 8Take the opposite side of your circle and fold it so that the curved partjust touches the center and the bottom forms a perfect point.
What does this look like? cone
Crease this well.
![Page 124: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/124.jpg)
Activity; The Magic Circle
Step 8Take the opposite side of your circle and fold it so that the curved partjust touches the center and the bottom forms a perfect point.
What does this look like?
cone
Crease this well.
![Page 125: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/125.jpg)
Activity; The Magic Circle
Step 8Take the opposite side of your circle and fold it so that the curved partjust touches the center and the bottom forms a perfect point.
What does this look like? cone
Crease this well.
![Page 126: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/126.jpg)
Activity; The Magic Circle
Step 8Take the opposite side of your circle and fold it so that the curved partjust touches the center and the bottom forms a perfect point.
What does this look like? cone
Crease this well.
![Page 127: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/127.jpg)
Activity; The Magic Circle
Step 9Fold the top of your ice cream cone down until the curved part justtouches the center of the circle. The top corners should make perfectpoints, crease well. Now describe the shape you have.
Do you notice anything special about this triangle?It is an equilateral triangle.
![Page 128: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/128.jpg)
Activity; The Magic Circle
Step 9Fold the top of your ice cream cone down until the curved part justtouches the center of the circle. The top corners should make perfectpoints, crease well. Now describe the shape you have.
Do you notice anything special about this triangle?
It is an equilateral triangle.
![Page 129: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/129.jpg)
Activity; The Magic Circle
Step 9Fold the top of your ice cream cone down until the curved part justtouches the center of the circle. The top corners should make perfectpoints, crease well. Now describe the shape you have.
Do you notice anything special about this triangle?It is an equilateral triangle.
![Page 130: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/130.jpg)
Activity; The Magic Circle
Step 10Fold the new triangle in half by matching up two of the points. Creasewell. The new crease splits the triangle in half. What is this linecalled?
altitude
Do you notice anything else about this triangle?It is a right triangle
![Page 131: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/131.jpg)
Activity; The Magic Circle
Step 10Fold the new triangle in half by matching up two of the points. Creasewell. The new crease splits the triangle in half. What is this linecalled? altitude
Do you notice anything else about this triangle?It is a right triangle
![Page 132: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/132.jpg)
Activity; The Magic Circle
Step 10Fold the new triangle in half by matching up two of the points. Creasewell. The new crease splits the triangle in half. What is this linecalled? altitude
Do you notice anything else about this triangle?
It is a right triangle
![Page 133: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/133.jpg)
Activity; The Magic Circle
Step 10Fold the new triangle in half by matching up two of the points. Creasewell. The new crease splits the triangle in half. What is this linecalled? altitude
Do you notice anything else about this triangle?It is a right triangle
![Page 134: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/134.jpg)
Activity; The Magic Circle
Step 11Open the right triangle up to the equilateral triangle.
Take the top corner of the big triangle and fold it along the crease ofthe height. You can match the top point up to the bottom crease line.On the inside you will now see three smaller triangles.
Turn the paper over so that you do not see the creases. What is thisshape called? trapezoid
![Page 135: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/135.jpg)
Activity; The Magic Circle
Step 11Open the right triangle up to the equilateral triangle.
Take the top corner of the big triangle and fold it along the crease ofthe height. You can match the top point up to the bottom crease line.On the inside you will now see three smaller triangles.
Turn the paper over so that you do not see the creases. What is thisshape called? trapezoid
![Page 136: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/136.jpg)
Activity; The Magic Circle
Step 11Open the right triangle up to the equilateral triangle.
Take the top corner of the big triangle and fold it along the crease ofthe height. You can match the top point up to the bottom crease line.On the inside you will now see three smaller triangles.
Turn the paper over so that you do not see the creases. What is thisshape called?
trapezoid
![Page 137: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/137.jpg)
Activity; The Magic Circle
Step 11Open the right triangle up to the equilateral triangle.
Take the top corner of the big triangle and fold it along the crease ofthe height. You can match the top point up to the bottom crease line.On the inside you will now see three smaller triangles.
Turn the paper over so that you do not see the creases. What is thisshape called? trapezoid
![Page 138: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/138.jpg)
Activity; The Magic Circle
Step 12Turn it back over so that you now see all of the creases. Fold one ofthe outer triangles in so that it lies directly on top of the centertriangle.
Turn it back over. What shape do yo see? rhombus
![Page 139: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/139.jpg)
Activity; The Magic Circle
Step 12Turn it back over so that you now see all of the creases. Fold one ofthe outer triangles in so that it lies directly on top of the centertriangle.
Turn it back over. What shape do yo see?
rhombus
![Page 140: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/140.jpg)
Activity; The Magic Circle
Step 12Turn it back over so that you now see all of the creases. Fold one ofthe outer triangles in so that it lies directly on top of the centertriangle.
Turn it back over. What shape do yo see? rhombus
![Page 141: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/141.jpg)
Activity; The Magic Circle
Step 13Turn your shape back over and fold the last outer triangle over ontothe center one again. You should now have a smaller equilateraltriangle.
![Page 142: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/142.jpg)
Activity; The Magic Circle
Step 14Open up all three of the small triangles. Bring the three loose pointstogether.
What shape do you have now? pyramid
Note: this would be an opportunity to discuss bases, faces,verticesand edges.
![Page 143: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/143.jpg)
Activity; The Magic Circle
Step 14Open up all three of the small triangles. Bring the three loose pointstogether.
What shape do you have now?
pyramid
Note: this would be an opportunity to discuss bases, faces,verticesand edges.
![Page 144: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/144.jpg)
Activity; The Magic Circle
Step 14Open up all three of the small triangles. Bring the three loose pointstogether.
What shape do you have now? pyramid
Note: this would be an opportunity to discuss bases, faces,verticesand edges.
![Page 145: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/145.jpg)
Activity; The Magic Circle
Step 14Open up all three of the small triangles. Bring the three loose pointstogether.
What shape do you have now? pyramid
Note: this would be an opportunity to discuss bases, faces,verticesand edges.
![Page 146: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/146.jpg)
Activity; The Magic Circle
Step 15Open your pyramid back up to the large equilateral triangle.
Fold over one of the points so that it just touches the dot in themiddle. What shape have you re-created?
![Page 147: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/147.jpg)
Activity; The Magic Circle
Step 15Open your pyramid back up to the large equilateral triangle.
Fold over one of the points so that it just touches the dot in themiddle. What shape have you re-created?
![Page 148: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/148.jpg)
Activity; The Magic Circle
Step 16Fold one more of the points in so that it just touches the dot in themiddle. Now what shape do you have?
pentagon
Now fold in the last point. What shape is it now? hexagon
Note: this is a good place to discuss planar figures
![Page 149: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/149.jpg)
Activity; The Magic Circle
Step 16Fold one more of the points in so that it just touches the dot in themiddle. Now what shape do you have? pentagon
Now fold in the last point. What shape is it now? hexagon
Note: this is a good place to discuss planar figures
![Page 150: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/150.jpg)
Activity; The Magic Circle
Step 16Fold one more of the points in so that it just touches the dot in themiddle. Now what shape do you have? pentagon
Now fold in the last point. What shape is it now?
hexagon
Note: this is a good place to discuss planar figures
![Page 151: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/151.jpg)
Activity; The Magic Circle
Step 16Fold one more of the points in so that it just touches the dot in themiddle. Now what shape do you have? pentagon
Now fold in the last point. What shape is it now? hexagon
Note: this is a good place to discuss planar figures
![Page 152: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/152.jpg)
Activity; The Magic Circle
Step 16Fold one more of the points in so that it just touches the dot in themiddle. Now what shape do you have? pentagon
Now fold in the last point. What shape is it now? hexagon
Note: this is a good place to discuss planar figures
![Page 153: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/153.jpg)
Activity; The Magic Circle
Step 17Turn to the other side and fit one of the corners into a flap on theopposite side of the triangle. You may have to try more than one.Choose the one that makes the best fit. Slide the last cornerunder/inside the others.
What shape do you have now? truncated tetrahedron
![Page 154: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/154.jpg)
Activity; The Magic Circle
Step 17Turn to the other side and fit one of the corners into a flap on theopposite side of the triangle. You may have to try more than one.Choose the one that makes the best fit. Slide the last cornerunder/inside the others.
What shape do you have now?
truncated tetrahedron
![Page 155: Circlesbtravers.weebly.com/uploads/6/7/2/9/6729909/circles.pdf · Preliminaries Central Angles The measure of a minor arc is the same as the central angle cutting off that arc. Central](https://reader034.vdocument.in/reader034/viewer/2022042109/5e8938c1b76422254d2e0417/html5/thumbnails/155.jpg)
Activity; The Magic Circle
Step 17Turn to the other side and fit one of the corners into a flap on theopposite side of the triangle. You may have to try more than one.Choose the one that makes the best fit. Slide the last cornerunder/inside the others.
What shape do you have now? truncated tetrahedron