lesson 10.1
DESCRIPTION
Lesson 10.1. Today, we are going to… > identify segments and lines related to circles > use properties of tangents to a circle. Parts of a Circle. C. Circle C. Diameter = _ radius . Y. N. BN. YX. AB. A. C. X. B. A chord is. Y. YX. AB. A. C. X. B. A secant is. AB. - PowerPoint PPT PresentationTRANSCRIPT
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Lesson 10.1Parts of a Circle
Today, we are going to…> identify segments and lines related
to circles> use properties of tangents to a circle
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C
Circle C
Diameter = _ radius
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C
A chord is
YX
AB
A
B
X
YN
BN
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C
A secant is
A
B
X
Y
YX
AB
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C
A tangent is
ABA
B
Y X
XY
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internal tangents
Common Tangent Lines
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external tangents
Common Tangent Lines
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Two circles can intersect in 2, 1, or 0 points.
Draw 2 circles that have2 points of intersection
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internally tangent circles
Draw two circles that have1 point of intersection
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externally tangent circles
Draw two circles that have1 point of intersection
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concentric circles
Draw two circles that have no point of intersection
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9. What are the center and radius of circle A?
Center: Radius =
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10. What are the center and radius of circle B?
Center: Radius =
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11. Identify the intersection of the two circles.
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12. Identify all common tangents of the two circles.
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mABC =
A
B
C
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Theorem 10.1 & 10.2A line is tangent to a circle if
and only if it is _____________ to the radius from the point of
tangency.
A
B
C
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7
13. Find CA.
15D
C
B
AWhat is DA?
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7
14. Find x.
15
x
6
C
B
A
xx
168
What is CA?
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7
156
C
B
A
2610
24
How do we test if 3 segments create a right triangle?
15. Is AB a tangent?
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7
156
C
B
A
178
12
16. Is AB a tangent?
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17. Find the slope of line t.
A
C
A (3,0) and C (5, -1)
Slope of AC?
Slope of line t?
t
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C
A tangent segment
A B
One endpoint is the point of tangency.
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Theorem 10.3If 2 segments from the same
point outside a circle are tangent to the circle, then
they are congruent.
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7x - 2
3x + 8
18. Find x.
A
C
B
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x2 + 25
50
19. Find x.
A
C
B
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Lesson 10.2Arcs and Chords
Today, we are going to…> use properties of arcs and chords
of circles
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C
An angle whose vertex is the center of a circle is a
central angle.
A
B
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C
Minor Arc - Major Arc
A
B
D
Minor Arc
AB
Major Arc
ADB
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C A
B
D
60˚
m AB =
Measures of Arcs
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C
Semicircle
m AED = m ABD = m AD
A
B
D
E
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Find the measures of the arcs.
1. m BD
2. m DE
3. m FC
4. m BFD
D
E
F
B
C
100˚52˚
68˚
53˚?
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AD and EB are diameters.
F
A
B
D
E
C
5. Find x, y, and z.
30˚
z˚
x˚
y˚
x =
y =
z =
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Theorem 10.4
Two arcs are congruent if and only if their chords
are congruent.
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(2x + 48)°(3x + 11)°
B
ADC
6. Find m AB
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Theorem 10.5 & 10.6
A chord is a diameter if and only if it is a
perpendicular bisector of a chord and bisects its arc.
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7. Is AB a diameter?A
B
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8. Is AB a diameter?A
B
8
8
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9. Is AB a diameter?A
B
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Theorem 10.7
Two chords are congruent if and only if they are equidistant from the
center.
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AB = 12
10. Find CG.
DE = 12
7D
G BA
C
F
E6
x
?
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Lesson 10.3Inscribed Angles
Today, we are ALSO going to…> use properties of inscribed angles
to solve problems
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An inscribed angle is an angle whose vertex is on the
circle and whose sides contain chords of the circle.
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Theorem 10.8If an angle is inscribed,
then its measure is half the measure of its intercepted
arc.
x2x
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1. Find x.
x°
120°
x = 60°
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2. Find x.
x°
70°
x = 140°
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Theorem 10.9If 2 inscribed angles
intercept the same arc, then the angles are
congruent.
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3. Find x and y.
y°
45°
x°
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InscribedPentagon
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x°
A
D
C
B
4. DC is a diameter. Find x.
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Theorem 10.10If a right triangle is inscribed in a circle, then the hypotenuse is a
diameter of the circle.
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5. Find the values of x and y.
x°
y°A
42 D
C
B
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Theorem 10.11If a quadrilateral is inscribed in a
circle, then its opposite angles are
supplementary.
21
4 3
m 1 + m 3 = 180º
m 2 + m 4 = 180º
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6. Find the values of x and y.
x°
110°
80° y°
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7. Find the values of x and y.
x°
120°
100° y°
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Lesson 10.4Angle Relationships
in CirclesToday, we are going to…> use angles formed by tangents and
chords to solve problems > use angles formed by intersecting
lines to solve problems
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Theorem 10.12
If a tangent and a chord intersect at a point on a
circle, then...
GSP
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Theorem 10.12
… the measure of each angle formed is half the measure of its
intercepted arc.
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1A
BC
2
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1A
BC
2
1. Find m 1 and m 2.
100°
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2. Find and mACB and mAB
95°A
B
C
![Page 66: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/66.jpg)
3. Find x
5x°A
B
C(9x + 20)˚
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Theorem 10.13If 2 chords intersect inside a circle, then…
A
B
C
D
1
![Page 68: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/68.jpg)
B
CA
D
1
…the measure of the angle is half the sum of the intercepted arcs.
![Page 69: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/69.jpg)
A
B
C
D
x°
4. Find x.100°
120°
![Page 70: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/70.jpg)
A
B
C
D
x°
5. Find x.130°
160°
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A
B
C
D
x°
6. Find x.
80° 90°y°
![Page 72: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/72.jpg)
A
B
C
D
x°7. Find x.
100°
120°
![Page 73: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/73.jpg)
A
B
C
D
x°
8. Find x.
52°74°
Do you notice a pattern?
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Theorem 10.14If a tangent and a secant, two tangents, or two secants intersect outside a circle, then…
A
C
D
1
![Page 75: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/75.jpg)
Theorem 10.14If a tangent and a secant, two tangents, or two secants intersect outside a circle, then…
A
B
C 1
![Page 76: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/76.jpg)
Theorem 10.14If a tangent and a secant, two tangents, or two secants intersect outside a circle, then…
A
BC
D
1
![Page 77: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/77.jpg)
A
BC
D1
…the measure of the angle is half the difference of the intercepted arcs.
![Page 78: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/78.jpg)
9. Find x.
20° 80°
A
BC
D
x°
![Page 79: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/79.jpg)
10. Find x.
24°90°
A
BC
Dx°
![Page 80: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/80.jpg)
11. Find x.
200°x°
![Page 81: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/81.jpg)
A
C
D
12. Find x.
135°x°
![Page 82: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/82.jpg)
13. Find x.
100°
3 100°2 1
100°
60°
![Page 83: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/83.jpg)
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Lesson 10.5Segment Lengths
in CirclesToday, we are going to…> find the lengths of segments of chords, tangents, and secants
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Theorem 10.15
If 2 chords intersect inside a circle, then the product of their “segments” are
equal.
![Page 86: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/86.jpg)
a · b = c · d
a
b
c d
![Page 87: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/87.jpg)
1. Find x.
6
8 4
x
![Page 88: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/88.jpg)
2. Find x.
3x
182x
3
![Page 89: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/89.jpg)
3. Find x.
2x
18x
4
![Page 90: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/90.jpg)
Theorem 10.16 If 2 secant segments share the same endpoint outside
a circle, then…
GSP
![Page 91: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/91.jpg)
…one secant segment times its external part
equals the other secant segment times its external part.
![Page 92: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/92.jpg)
a · c = b · d
b
a
c
d
![Page 93: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/93.jpg)
3. Find x.
5
x
4 6
![Page 94: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/94.jpg)
4. Find x.
9
10x
20
![Page 95: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/95.jpg)
Theorem 10.17 If a secant segment and a tangent segment share an endpoint outside a circle,
then…
![Page 96: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/96.jpg)
…the length of the tangent segment squared equals the
length of the secant segment times its external
part.
![Page 97: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/97.jpg)
a · a = b · d
db
a
a2 = b · d
![Page 98: Lesson 10.1](https://reader036.vdocument.in/reader036/viewer/2022062222/56815cca550346895dcad60c/html5/thumbnails/98.jpg)
54
x5. Find x.
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15x
106. Find x.
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Quadratic Formula?♫♪♫♪♫♪♫♪♫♪♫♪
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15x
106. Find x.
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x20
317. Find x.
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8. Find x.
3
48
x
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10x
89. Find x.
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Lesson 10.6Equations of
CirclesToday, we are going to…> write the equation of a circle
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Standard Equation for a Circle with
Center: (0,0) Radius = r
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1. Write an equation of the circle.
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2. Write an equation of the circle.
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Standard Equation for a Circle with
Center: (h,k) Radius = r
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3.Write an equation of the circle.
C =
r =
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4.Write an equation of the circle.
C = r =
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Graph (x – 3)2 + (y + 2)2 = 9
Center?
Radius =
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Identify the center and radius of the circle with the given equation.
5. (x – 1)2 + (y + 3)2 = 100
6. x2 + (y - 7)2 = 8
7. (x + 1)2 + y2 = ¼
Center: (1, -3) radius = 10
Center: (0, 7) radius ≈ 2.83
Center: (-1, 0) radius = ½
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Write the standard equation of the circle with a center of (5, -1) if a point on the circle is (1,2).
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8. Write the standard equation of the circle with a center of (-3, 4) if a point on the circle is (2,-5).
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Is (-2,-10) on the circle (x + 5)2 + (y + 6)2 = 25?
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9. Is (0, - 6) on the circle (x + 5)2 + (y – 5)2 = 169?
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10. Is (2, 5) on the circle (x – 7)2 + (y + 5)2 = 121?
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><
=
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Would the point be inside the circle, outside the circle, or on the circle?
(x – 13)2 + (y - 4)2 = 100
11. (11, 13)
12. (6, -5)
13. (19, - 4)
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Lessons 11.4 & 11.5Circumference and
Area of CirclesToday, we are going to…> find the length around part of a circle and find the area of part of a circle
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Circumference
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Arc Length
=
A
B
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A
B50°7 cm
1. Find the length of AB
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A
B
85°
10 cm
2. Find the radius
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3. Find the circumference.
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Area
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Sector of a circle A region bound by two radii &
their intercepted arc.
A slice of pizza!
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Area of a Sector
=
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3. Find the area of the sector.A
B50°7 cm
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4. Find the radius. A
B
100°
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3. Find the area.
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Workbook
P. 211 (1 – 10)
P. 215 (1 – 6)
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