THEOREM: If two secants intersect outside a circle, then the product of the measures of one secant segment and its external segment is equal to the product of the measures of the other secant segment and its external segment.

Proof of theorem:Given: PB and PD are secants drawn from external point P.Prove: ΔPCB ~ ΔPAD

B

A

DC

P

Reasons

Aim #11: What theorems relate to lengths of secants and tangents in circles?Do Now: How many common tangent lines can be drawn to the circles shown below?

CC Geometry H

Statements

a

cb

d

a(a + b) = c(c + d)

Write a proportion that would be true and rearrange it to prove that the theorem above is true.

1) Find:

a) m≮a:

b) m≮b:

c) Find x:

2) a) Find x: b) If OA = 5, EB = 8, EC = 9, find ED:

2515

5x

O

B

C

D

E

THEOREM: If a tangent and a secant are drawn to a circle from an external point, then the square of the measure of the tangent segment is equal to the product of the measures of the whole secant segment and its external segment.

A

a

cb

.

.a2 = b(b + c)

3) Find x. Lines that appear to be tangent are tangent:a) b)

12x

8

Q

PR

S

A 4 P

B6

C

Ox

c) d)

4) Find x:a) b)

c) d) CE = 6, CB = 9, and CD = 18.Show CF = 3.

5) PQ is tangent to circle O. PA = 9, PQ = 6. Find a) PB

b) the radius of circle O

PB

QO A

Mixed Practice. Find the value of each variable.

6) Find z:

16

x

8

Mixed Practice. Find the value of each variable. Lines that appear to be tangent are tangent:

93

x

46

x10

9 6

x

8

13

3

x

x8

12

17 4

5x

7) Find x to the nearest tenth. 8) 9)

2) 3)1)

4) 5) 6)

1. In the diagram ST = 7, TU = 1 and SK = 4. Find KM.

M

K

STU

17

4

x

2. PA is a tangent; CF = 4, FB = 3, BP = 5. AF : FE = 1 : 3. DE is 7 more than PD. Find AF, AP, EP, and PD.

4 3 5B

D

E

C

AP

F

P B

Q

O A

3. PQ is tangent to circle O. If the radius of the circle is 5 and PQ = 12, find PB.

4. The lengths of two secants drawn to a circle from an external point are 16 and 20. If the external segment of the shorter secant has length 5, find the external segment of the longer secant.

5. Two secants are drawn to a circle from an external point. One secant has length 8, and its external segment has length 3. The length of the internal segment of the other secant is 5 times the length of its external segment. Find the length of the second secant.

Name: _____________________ CC Geometry H Date: ____________ HW #11

Review:

1) As shown below, a canoe is approaching a lighthouse on the coastline of a lake.

The front of the canoe is 1.5 feet above the water and an observer in the lighthouse is 112 feet above the water.

At 5:00, the observer in the lighthouse measured the angle of depression to the front of the canoe to be 6°. Five minutes later, the observer measured and saw the angle of depression to the front of the canoe had increased by 49°. Determine and state, to the nearest foot per minute, the average speed at which the canoe traveled toward the lighthouse.

2) In right ΔABC with the right ≮ at C, sin A = 2x + 0.1 and cos B = 4x – 0.7. Determine and state the value of x. Explain your answer.

3) A parallelogram must be a square if the diagonals are:a. congruent and bisect the angles to which they are drawnb. congruent and do not bisect the angles to which they are drawnc. not congruent and bisect the angles to which they are drawnd. not congruent and do not bisect the angles to which they are drawn