Review Ch. 10

Complete all problems on a separate sheet of paper. Be sure to number each problem. Good Luck!

Problem #1

A

Find the exact length of arc AB, if circle

P has a radius of 18cm.

100°

B

P

Solution to #1

Arc Length = (100/360) * 36π

= (5/18) * 36π

= 10π cm.

Problem # 2

Find the diameter of a circle in which a 36 cm chord is 80 cm

from the center.

Solution to #2

This is a 9, 40, 41 triangle times 2 so r = 82cm diameter = 164 cm.

r

18cm18cm

80cm

Problem #3

20

Find the radius of a circle with a circumference of

Solution to #3

Circumference = π * diameter so the

diameter must be 20 so radius = 10.

Problem #4

Find the measure of arc AE.

210оx

200оA

B

C

DE

Solution to #4

*Arc BC = 360 – 210 = 150о

*Angle BDC is supp (tangent-tangent) = 30о

*So Angle ADE = 30о

*So 30 = (1/2)(200 – x)

60 = 200 – x

x = 140о

Problem #5

In the circumscribed polygon, find the length of the AB.

10

12

15 A

B

Solution to #5

AB = 15 – (10 – x) + 12 – x

= 5 + x + 12 – x

= 17

B

A

12 - x

12 - x

15 - (10 - x)

15 - (10 - x)

10 - x

10 - x

x

x

Problem #6

In circle O, AB is a diameter.

OA=3x+5 and OB=2(5x-1). Find AB.

Solution to #6

OA and OB are both radii so are equal.

3x + 5 = 2(5x – 1)

3x + 5 = 10x – 2

7 = 7x

1 = x

each radius = 8 ; so diameter AB = 16

Problem #7

Solve for x if

and if

A

B

C

5 6m A x 12 2mBC x

Solution to #7

Since angle A is inscribed;

2(5x + 6) = 12x – 2

10x + 12 = 12x – 2

14 = 2x

x = 7

Problem #8

MATH is inscribed in the circle.

Angle M has a measure of 78 degrees.

Find the measure of angle T.

M

A

H

T

Solution to #8

Opp. Angles of inscribed quadrilaterals are

supp.

Measure of Angle T = 180 – 78 = 102о

Problem #9

Find the radius of the circle if AB is a diameter, , and BC=20. 120mAC

A

B

C

Solution to #9

*Measure of Angle B = 120/2 = 60

*Measure of Angle C = 90

*30 – 60 – 90 triangle with x = 20 ; so

diameter is 2x = 40

*Radius of AB is 20.

30

20cm60

120

C

B

A

Problem #10A circle is inscribed in triangle ABC. AB=14, AC=12 and BC=4. Find BD.

A

B CD

Solution to #10

14 – x + 12 – x = 4

26 – 2x = 4

22 = 2x

x = 11

So BD = 14 – x = 312 - x

12 - x

14 - x

14 - x

xx

CDB

A

Problem #11

A circle has a radius of 50. How far from the center is a

chord of length 28?

Solution to #11

7, 24, 25 right triangle

So x = 2 * 24 = 48

x

14 14

50

Problem #12

A regular octagon is inscribed in a circle.

What is the measure of an arc cut off by a side of the

octagon?

Solution to #12

* Regular - so all chords congruent.

* Congruent chords = congruent

arcs.

360/8 = 45о

Problem #13Two concentric circles have radii of

lengths 16 and 20. Find the length of a chord of the larger circle that is

tangent to the smaller circle.

Solution to #13

• 3, 4, 5 right triangle

x = 12 so length of the chord

is 24.

x

20

16

Problem #14

A 12 by 10 rectangle is inscribed in a circle. Find the radius.

Solution to #14

• 144 + 100 = c2

244 = c2

10

12244 = c2 61 = diameterso radius = 61

Problem #15Two secants drawn to a circle from an external point intercept arcs that are 122° and 68°. Find the measure of the secant-

secant angle.

P

122°

68°

Solution to #15

• Angle P = (1/2)(122 – 68)

= (1/2)(54)

= 27о

Problem #16

Find the circumference of a circle in which an 80 cm chord

is 9 cm from the center.

Solution to #16

• 9, 40, 41 right triangle so r = 41

• C = 2π(41)

= 82 π cm r

9

40

40

Problem #17A central angle intercepts an arc that is 5/12

of the circle. Find the measure of angle x.

512

of circle O

O x

Solution to #17

• If arc is 5/12 central angle is 5/12 of 360 so central angle is 150о

• Radii are congruent so isosceles triangle only 30о left.

• Angle x = 30/2 = 15о

Problem #18If PA and PB are tangent to circle O at A

and B, PA=24, and PO=26, find

perimeter of quadrilateral PAOB.

P

A

B

O

Solution to #18

• OA is perpendicular to PA 5, 12, 13 right triangle.

• OA = 10 and PB = 24

• 10 + 10 + 24 + 24 = 68

Problem #19

Find the measure of angle x.

x

92°44°

Solution to #19

x = 180 - 68x = 112

? = (1/2)(44 + 92)? = (1/2)(136)? = 68

?x

44

92

Problem #20

What is the length of a chord that cuts off an arc of 120 degrees in a circle with a radius of 8?

Solution to #20

x 3

30, 60, 90 triangle 2x = 8 x = 4

Chord = 2(4 3) = 8 3

8

60

Problem #21

Parallelogram ABCD is inscribed in circle Q, with dimensions of 24 by 10. Find the area of circle Q.

Solution to #21

Parallelogram inscribedis a rectangle.

Diameter = 26 radius = 13

Area = (132) = 169

10

24

Problem #22

AB

C

•

•

•

•D

Circle A has a radius of 5 inches, and circle B has a radius of 20 inches. The centers are 39 inches apart. Find the length of the common external tangent (CD).

Solution to #22

E

Rt. Triangle is 5, 12, 13 soAE = 12(3) = 36 in. Sothen CD = 36 in.

15 in

5 in 5 in

39 in.

D C

A

B

Problem #23

Two tangent segments of a circle with a diameter of 50 inches form a 60 degree angle where they meet at P. How far is

P from the center of the circle?

60°P

Solution to #23

30 6025 in

If P = 60 , then mAB = 120 ,so ACB = 120 . PC bisectsACB so ACP = 60 . Wehave a 30-60-90 triangle withx = 25 in.

PC = 50in.

B

C

A

P

Problem #24

AB & AC are tangent to the circle.

Find the measure of arc BDC.

A

B

C

D

76°

Solution to #24Minor arc supp. to angle = 104

mBDC = 360 - 104 = 256

10476

C

D

B

A

STUDY!!!!!