9-1 basic terms associated with circles and spheres
TRANSCRIPT
Geo 9 1 Circles 9-1 Basic Terms associated with Circles and Spheres Circle __________________________________________________________________ Given Point = __________________ Given distance = _____________________ Radius__________________________________________________________________ Chord____________________________________________________________________ Secant___________________________________________________________________ Diameter__________________________________________________________________ Tangent___________________________________________________________________ Point of Tangency___________________________________________________________ Sphere____________________________________________________________________ Label Accordingly: Congruent circles or spheres__________________________________________________ Concentric Circles___________________________________________________________ Concentric Spheres__________________________________________________________ Inscribed in a circle/circumscribed about the polygon________________________________ _______________________________________ http://www.pinkmonkey.com/studyguides/subjects/geometry/chap7/g0707101.asp
Geo 9 3 Circles 9-2 Tangents POWERPOINT
Theorem 9-1 If a line is tangent to a circle , then the line is __________________________
_________________________________.
Corollary: Tangents to a circle from a point are __________________________
Theorem 9-2 If a line in the plane of a circle is perpendicular to a radius at its outer endpoint, then
the line is ________________________.
Inscribed in the polygon/circumscribed about the circle:
P
A
B
look for 2 tangents from the same point!
what if A is a right ange?
A
Geo 9 4 Circles Common Tangent ___________________________________________________ Common Internal Tangent Common External Tangent Tangent circles ________________________________________________________ Draw the tangent line for each drawing Name a line that satisfies the given description. 1. Tangent to P but not to O. _______ 2. Common external tangent to O and P. _______ 3. Common internal tangent to O and P. _______
A B
O
P
C F
Geo 9 5 Circles 4. Circles A, B, C are tangent . AB = 7, AC = 5 CB = 9 Find the radii of the circles. 5. Find the radius of the circle inscribed in a 3-4-5 triangle. PP CONCLUSION
A
B
C
x
3
5
4
Geo 9 6 Circles
6) Circles O and P have radii 18 and 8 respectively. AB is tangent to both circles. Find AB…………….Hint: connect centers. Find a rt.
A
B
P
O
Geo 9 7 Circles 9-3 Arcs and Central Angles Central Angle ________________________________________________________
Arc ________________________________________________________________
Measure of a minor arc = ______________
Measure of a major arc = __________ - ______________
Adjacent arcs ____________________
Measure of a semicircle = ___________________
Postulate 16 Arc Addition Postulate: The measure of the arc formed by two adjacent arcs is
_________________________________________.
That is, arcs are additive. Just like with angles, to differentiate an arc from its measure, an “m” must be included in front of the arc. Congruent arcs _______________________________
Theorem 9-3 In the same circle or _________________, two minor arcs are _____________ if
_________________________________.
1. Name 2. Give the measure of each angle or arc:
a) two minor arcs a) AC
b) two major arcs b) m WOT
c) a semicircle c) XYT
d) an acute central angle
e) two congruent arcs
R
S
O
X
Y
Z
T
W
O
30
50
A C
Geo 9 8 Circles
3. Find the measure of 1 (the central angle) a) b) c) d) 4. Find the measure of each arc: a) AB b) BC c) CD d) DE e) EA
5) a) If 60CB , AO = 10, find <1, <2 and AB
b) If <2 = x find <1, CB
2x-14
2x
3x+10
3x
4x
A B
C
D
E
A
B
C O
1 2
1
130
1
72
1
40
225
30
1
Geo 9 9 Circles 9-4 Arcs and Chords The arc of the chord is _______________________________________
Theorem 9-5 A diameter that is perpendicular to a chord _______________ the chord and
_________________________.
That is, in O with CD AB, AZ = BZ and AD BD How?
Other Theorems: If < AOB = < COD, then what must be true as well? 1) 2) 3) 4)
D
O
C
B
A
A B
O
Z
C
D
Geo 9 10 Circles Find the following: 1. x = ______ y = ______ 2. x = ______ y = ______ mAB = ______
3. MN = ______ KO = ______ 4. = ______ m AOC = ______
5. x = ______ y = ______ 6. mCD = ______
7. CD = 40 , FIND CA 8. If OC = 6, find x and y
x y
5
13
C O
A
B
220
M
K
15
17
S
N
O
80
O
C
D 8
x
y
A B
C D
40A
A
C
60
D
B
O
x y
E
6
x
y 6 60
B A
ACB
D
Geo 9 11 Circles
9-5 Inscribed Angles
By definition, an inscribed angle is an angle whose VERTEX IS ON THE CIRCLE and is
contained in the circle. Inscribed angles can intercept a minor arc or a major arc.
Theorem 9-7 The measure of an inscribed angle is equal to ________________________________
Find angle A and angle B. What generalization can you make?
Corollary 1: If two inscribed angles __________________ _____________________________
Corollary 2: An inscribed angle that intercepts a diameter _________________________________
D C
A
B
70
Geo 9 12 Circles
Corollary 3: If a quadrilateral is inscribed in a circle, then its opposite angles are ________________
Theorem 9-8 the measure of an angle formed by a chord and a tangent is equal to
____________ of the intercepted ___________.
Solve for the variable(s) listed:
80
z
x
y 60
x
y
z
80
x
y
A
B
C
D
X
Y
Geo 9 13 Circles POWERPOINT
60
x
140
y
x
110
20
y
20
x
y
50
Geo 9 14 Circles 9-6 Other Angles Sketchpad
Theorem 9-9 The measure of an angle formed by two chords that intersect inside a circle is equal to
1
2 the sum of the intercepted arcs.
That is: ____________________
Theorem 9-10 The measure of an angle formed by secants, two tangents
or a secant and a tangent is equal to ______________________________________
THE VERTEX IS OUTSIDE THE CIRCLE Case 1 Case 2 Case 3 2 secants 2 tangents secant/tangent _________________ _________________ __________________
y x
x
y x
y
x
y
1
Geo 9 15 Circles
Given UT is tangent to the circle, m VUT = 30. Find the following:
1. m WT = ________ 2. m TVS = ________ 3. m RVS = ________ 4. m RS = ________
Given the drawing: AB is tangent to O; AF is a diameter; m AG = 100, mCE = 30,
m EF = 25. Find the measures of angles 1-8. 1= 2= 3= 4= 5= 6= 7= 8=
F
G
1 2
3
4
5
6
7
C
E
A
B
8 O
R
100
S
V
T
U
100
W
Geo 9 16 Circles
ANGLE MEASUREMENT BASED ON VERTEX
1) VERTEX AT CENTER angle = ______________
2) VERTEX ON CIRCLE angle = ______________
3) VERTEX INSIDE CIRCLE angle = ______________
4) VERTEX OUTSIDE THE CIRCLE angle = ______________
SECANT/SECANT TANGENT/SECANT TANGENT/TANGENT
2 1
1 2
1 2
Geo 9 17 Circles 9-7 Circles and Lengths of Segments
Theorem 9-11 When two ________ intersect inside a circle, the __________ of the _______
of _______ ____________ equals the ___________ of the ______________
of the ___________ ______________.
That is, in the circle below, given that the two chords intersect, the equation is ____________ or __________________________
Theorem 9-12 When two ________ segments are drawn to a circle from an _________
_____________, the product of one secant segment and its __________
______________ is equal to the product of the other secant segment and
its _______________________
That is, in the circle below,
_____________ or _______________________________
Theorem 9-13 When a _______ segment and a _________ segment are drawn to a circle
From an ___________ ________ the product of the secant segment and
Its _______ _________ is equal to the __________ of the ____________.
That is, in the circle below: _______________ or ____________________________
r
s
t
t
s
r
u
t
s
r
u
Geo 9 18 Circles EXAMPLES:
SKETCHPAD POWERPOINT
x
10
12
3
15
4
x
x
x
y
4 5
9
y
1
3 3
x
2x
2
y
4
2
x 4
5
7
4
12
18
x
x
y
3 10
5
4
6
Geo 9 19 Circles Find the measure of each numbered angle given arc measures as indicated.
42 is a central angle
m 1__________ m 2__________ m 3__________ m 4___________ m 5___________
m 6__________ m 7__________ m 8__________ m 9___________ m 10__________
m 11_________ m 12_________ m 13_________ m 14__________ m 15__________
m 16_________ m 17_________ m 18_________ m 19__________ m 20__________
m 21_________ m 22_________ m 23_________ m 24__________ m 25__________
m 26_________ m 27_________ m 28_________ m 29__________ m 30__________
m 31_________ m 32_________ m 33_________ m 34__________ m 35__________
m 36_________ m 37_________ m 38_________ m 39__________ m 40__________
m 41_________ m 42_________ m 43_________ m 44__________ m 45__________
8 9
10
11 12
13 43
44 45
35
15
16 17 18 19
20
22
23
24
25
26
27
29 30
31
32
33
34
36
37
38
39 40
41
1
2
3
4 5
6
14
45 21
28
35
42
7
40
60
50
20
20
Geo 9 20 Circles
CH 9 CIRCLE REVIEW (1) Find the measure of each of the numbered (2) The three circles with centers A , B , and C angles, given the figure below with arc are tangent to each other as shown below. measures as marked. Point O is the center Find the radius of each circle if AB = 12 , of the circle. AC = 10 and BC = 8.
m 1 =____ m 2 =____ m 3 =____ m 4 =____
m 5 =____ m 6 =____ m 7 =____ m 8 =____ Circle A_____ , Circle B_____ , Circle C_____
m 9 =____ m 10 =____
(3) mAB= 120 , AO = 6. Find: AB_____ (4) m A = 80 Find: mBDC______
(5) BC is tangent to the circle with center O. (6) AB is a diameter, CD AB , AC = 3 , AB = 2 , OC = 3. Find: BC______ BC = 6. Find: CD______
O
60
5 6
7
8
9
10 140
50
1 2 3
4 40
O
A
B
120
6
B
A
C
O A
B
C
D 80
3
O
A
B
C
2
3 6 A B C
D
Geo 9 21 Circles
(7) AE is tangent at B, CD is a diameter, (8) AB is a diameter, BC is tangent at B,
m A = 40 . Find: mBD ____, m EBD____ mAD = 120 , AD = 36 .
Find: BC_____, CD_____, OA_____
(9) AB is tangent at A, AF = FD, sides as marked. (10) Given the figure with sides as marked, Find: EF______ , AF_______ Find: BC_______ , EF_______ (11) Circles with centers O and P as shown, (12) Given the figure below with sides as OP = 15 , OC = 8 , PD = 4 marked, find the radius of the inscribed Find: AB______ , CD_______ circle________
36
O A B
C
D 120
O A
B
40
C D
E
O
B
A
C
D
P
16
12
20
O
A
B C
D
E
F
A
C
D
E
F B
4 3
34 14
D
B A
C
E 6
6
10 5
4
F
Geo 9 22 Circles
Answers
(1) m 1 = 20 , m 2 = 25 , m 3 = 55 , m 4 = 90
m 5 = 25 , m 6 = 115 , m 7 = 65 , m 8 = 115
m 9 = 45 , m 10 = 130
(2) Circle A = 7 , Circle B = 5 , Circle C = 3
(3) 6 3
(4) mBDC = 260 (5) BC = 4
(6) CD = 3 2
(7) mBD = 130 , m EBD = 65
(8) BC = 4 3 , CD = 2 3 , OA = 6
(9) EF = 9 , AF = 6 (10) BC = 4 , EF = 8
(11) AB = 9 , CD = 209
(12) 4
Geo 9 23 Circles
CH 9 CIRCLES REVIEW II
(1) The circle with center O is inscribed in ABC. (2) CA is tangent to the circle at A,
sides as
AC BC . Find: AC______ , BC_______ marked. Find: AC_______
(3) AB is an external tangent segment. Points (4) Concentric circles with center O, AC is
O and P are the centers of the circles. tangent to the inner circle, sides as marked.
Find: AB_________ Find: OB_______ , mADC________
(5) Given the figure below, point O is the center (6) Given the figure below, m A = 30 ,
the circle, AC BD , BD = 26 , AC = 24. m CFD = 65 , BC = DE.
Find: OE_____ , DE_____ , OC______ Find: mCD____, mBE ____, mBC____
6
4
O
A
B C
D
E
F
D
P
A
O
C
B
4 6
30 65 A
B
C
D
E
F
O
A
B
C
D
E
6
O
A
B
C
6
8
38
O
A B C
D
Geo 9 24 Circles (7) The circle below with center O, AC = 12 , (8) Given the figure below, DH = HF, with
AC BD . sides as marked.
Find: OE______ , OC_______DE_______ Find: GC_______ , DH________
(9) The circle with center O is inscribed (10) Points O and P are the centers of the
in ABC as shown below. AB = AC, circles below. CP = 6
sides as marked. Find: OE_________ Find: AB_______ , mACB________
(11) A chord whose length is 30 is in a circle whose radius is 17. How far is the chord from the
center of the circle?
B
120
O
A C
D
E
A
B C D
E
F
G H
3
4 6
3
B
8
5
C
O
A
D E
F
6
O P
A
B
C
Geo 9 25 Circles
Review Answers II
(1) AC = 6 , BC = 8
(2) AC = 6 3
(3) AB = 4 6
(4) OB = 4 , mADC = 240 (5) OE = 5 , DE = 8 , OC = 13
(6) mCD = 95 , mBE = 35 , mBC = 115
(7) OE = 2 3 , OC = 4 3 , DE = 2 3
(8) GC = 27
4 , DH = 3 3
(9) OE = 10
3
(10) AB = 6 3 , mACB = 240
(11) 8
Geo 9 26 Circles
CH 9 CIRCLES ADDITIONAL REVIEW 1) Find the radius of a circle in which a 48 cm chord is 8 cm closer to the center than a 40 cm chord. AB = 48, CD = 40 2) In a circle O, PQ = 4 RQ = 10 PO = 15. Find PS.
3) An isosceles triangle, with legs = 13, is inscribed in a circle. If the altitude to the base of the triangle is = 5, find the radius of the circle. (There are 2 situations) Answers: 1) 25 2) 2 3) 16.9
A B
C D
R
S O
P Q
13 13
13 13
Geo 9 27 Circles
SUPPLEMENTARY PROBLEMS CH 9
1) Fill out page one of the Circles Packet. 2) A regular polygon is inscribed in a circle so that all vertices of the quadrilateral intersect the circle. What happens to the regular polygon as the number of sides increases. 3) A circle with a center at (2,1) is tangent to the line y = 3x + 5 at A(-1,2). Make a sketch in the coordinate plane and draw a radius from the center of the circle to the radius at point A? Why? 4) In the picture below, AB is a common external tangent. How many common external tangents can be drawn connecting the 2 circles in each of the following pictures? What shape can be formed if a radius drawn to a tangent is perpendicular to the tangent?
5) If the central angle of a slice of pizza is 36 degrees, how many pieces are in the pizza? 6) Circle O has a diameter DG and central angles COG = 86, DOE = 25, and FOG = 15. Find the minor arcs CG, CF, EF, and major arc DGF. 7) Draw a circle and label one of its diameters AB. Choose any other point on the circle and call it C. What can you say about the size of angle ACB? Does it depend on which C you chose? Justify your response, please.
A
B
9.2
TA
NG
EN
TS
9
.3 A
RC
S A
ND
CE
NT
RA
L A
NG
LE
S
Geo 9 28 Circles 8) If two chords in the same circle have the same length, then their minor arcs have the same length, too. True or false? Explain. What about the converse of the statement? Is it true? Why? 9) Draw a circle. Draw two chords of unequal length. Which chord is closer to the center of the circle? What can be said of the “intercepted arcs”? 10) If P and Q are points on a circle, then the center of the circle must be on the perpendicular bisector of chord PQ. Explain. Which point on the chord is closest to the center? 11) The Star Trek Theorem: a.) Given a circle centered at O, let A,B,and C be points on the circle such that arc AC is not equal to arc BC and CL is a diameter. Why must triangles AOC and AOB be isosceles? b) State the pairs of angles that must be congruent in these isosceles triangles. c) Using EAT, find expressions for the measures of <AOL and <BOL. d) Based on your statement in part c, explain the statement <ACL = ½(<AOL) and <OCB = ½(<BOL). e) Now find an expression for <ACB and simplify to prove that it equals ½<AOB.
D
Q
P
9.4
AR
CS
AN
D C
HO
RD
S
B
L
C
O
A
9.5
IN
SC
RIB
ED
AN
GLE
S