GeometryREVISED SENT13 and 14

Angle-Arc Summary Central Angle Chord-Chord Angle

:0 ~ ( 15;-

___a.ogent Angle .i-> S

Secant-Tangent.

p~d B C

T 1 --------.....----.. 1 ~.....----.. 1 .....----..

mLP = 2(mCD - mAB) mLP = 2(mSXT - mST) mLP = 2(mRT - Vertex outside circle ~ half the difference

• Find y.

Find mLBEC first. mLBEC = ~(29 + 47) = 38 Thus, y = 180 - mLBEC = 142.

Part Two: Sample Problems Problem 1

104'

Given: AB is a diameter of OP. ~E ED = 20°, DE = 104° C 20° A

Find:mLC ~

First find mEA. ~ .....----..

mAEB = 180, so mEA = 180 - (104 + 20) = 56. 1""'----" .....----.. 1

Thus, mLC = 2(mEA - mDB) = 2(56 - 20) = 18.

Solution

Problem 2

Solution

472 Chapter 10 Circles

Problem 3

Solution

Problem 5

Solution

a Find x. b Find y.

Solution a x = ~(88 + 27) b y = ~(57 - 31) c z = ~(233 - 127)= 57.! = 13 = 532

Problem 4 a Find y. b Find z. c Find a.

Part Three: Problem Sets Problem Set A 1 Vertex at center:

Given: AB = 62° Find: mLO

a ~(21 + y) = 72 21 + Y = 144

Y = 123

b ~(125 - z) = 32 125 - z = 64

z = 61

Find mAB and mED. ../ Let mAB = x arid mED = y. Then ~(x + y) = 65 and ~(x - y) = 24. So x + Y = 130 and x - y = 48. x + Y = 130 x - Y = 48

2x = 178 Add the equations. X = 89

89 + Y = 130 Y = 41

Thus, mAB = 89 and mED = 41.

c Find z.

233

c ~a = 65 a = 130

F

Section 10.5 Angles Related to a Circle 473

Problem Set A, continued c

2 Vertex inside: Given: CD = 100°, Fe = 30° Find: mLCED

3 Vertex on: a Given: AC = 70°

Find: mLB

F B

Db Given: DE is tangent at E.EF = 150° Find: mLDEF

E 4 Vertex outside:

ba c w

R

K R

Given: Wand R are points of contact. WR = 140°

Find: mLX

T Given: HP = 120°,

AM = 36° Find: mLK

Given: TU is tangent at U. RD = 160°, §D = 60°

Find: mLT

5 Find the measure of each angle or arc that is labeled with a letter.

160'c ea

x

11

10

..---.... b d

12 1-, 120c 810

82c

474 Chapter 10 Circles

Problem 4 A walk-around problem: Given: Each side of quadrilateral

ABCD is tangent to the circle. AB = 10, BC = 15, AD = 18

Find: CD

Solution Let BE = x and "walk around" the figure, using the given information and the Two-Tangent Theorem. CD = 15 - x + 18 - (10 - x)

= 15 - x + 18 - 10 + x = 23

See problems 16, 21, 22, and 29 for other types of walk-around problems.

Part Three: Problem Sets Problem Set A

1 The radius of OA is 8 cm. Tangent segment BC is 15 cm long. Find the length of AC.

A

B x 15 - x ,c 15 - x

(10 - x)

10 x

A 10 - x

2 Concentric circles with radii 8 and 10 have center P. XY is a tangent to the inner circle and is a chord of the outer circle. Find XY. (Hint: Draw PX and PY.)

3 Given: PR and PQ are tangents to 00 at Rand Q.

-----7 _

Prove: PO bisects LRPQ. (Hint: Draw RO and OQ.)

4 Given: AC is a diameter of OB. Lines sand m are tangents to the o at A and C.

Conclusion: s II m

x

y

p~ --=-...'.R-,--

Q

Section 10.4 Secants and Tangents 463

Problem Set A, continued

5 OP and OR are internally tangent at O. P is at (8, 0) and R is at (19, 0). a Find the coordinates of Q and S. b Find the length of QR.

o

6 AB and AC are tangents to 00, and OC = 5x. Find OC.

B

A~ 19 - 6x C

7 Given: CE is a common internal tangent to circles A and B at C and E.

Prove: a LA == LB b AD = CD

BD DE

8 Given: QR and QS are tangent to OP at points Rand S.

Prove: PQ 1. RS (Hint: This can be proved in just a few steps.)

9 Given: PW and PZ are common tangents to @ A and B at W, X, Y, and Z.

Prove: WX == YZ (Hint: No auxiliary lines are needed.)

Note This is part of the proof of a useful property: The common external tangent segments of two circles are congruent.

Problem Set B 10 OP is tangent to each side of ABCD.

AB = 20, BC = 11, and DC = 14. Let AQ = x and find AD.

464 Chapter 10 Circles

w

p

z

A

x-axis

11 a Find the radius of OP. b Find the slope of the tangent to OP at

point Q.

x-axis

12 Two concentric circles have radii 3 and 7. Find, to the nearest hundredth, the length of a chord of the larger circle that is tangent to the smaller circle. (See problem 2 for a diagram.)

13 The centers of two circles of radii 10 em and 5 cm are 13 em apart. a Find the length of a common external tangent. (Hint: Use the

common -tangent procedure.) b Do the circles intersect?

14 The centers of two circles with radii 3 and 5 are 10 units apart. Find the length of a common internal tangent. (Hint: Use the common-tangent procedure.)

15 Given: PT is tangent to ® Q and R at points Sand T. . PQ SQ

Conclusion: PR = TR P -====-------i-*--l---+--------!

16 Given: Tangent ® A, B, and C, AB = 8, BC = 13, AC = 11

Find: The radii of the three ® (Hint: This is a walk-around problem.)

17 The radius of 00 is 10. The secant segment PX measures 21 and is 8 units from the center of the O. a Find the external part (PY) of the se-

cant segment. b Find OP.

T

P

Section 10.4 Secants and Tangents 465