10.5 Segment Lengths in Circles

GeometryMrs. Spitz

Spring 2005

Objectives/Assignment• Find the lengths of segments of

chords.

• Find the lengths of segments of tangents and secants.

• Assignment: pp. 632-633 #1-30 all.

• Practice Quiz pg. 635 #1-7 all.

Finding the Lengths of Chords

• When two chords intersect in the interior of a circle, each chord is divided into two segments which are called segments of a chord. The following theorem gives a relationship between the lengths of the four segments that are formed.

Theorem 10.15

• If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

E

A

B

C

D

EA • EB = EC • ED

Proving Theorem 10.15

• You can use similar triangles to prove Theorem 10.15.

• Given: , are chords that intersect at E.

• Prove: EA • EB = EC • ED

AB CD

E

D

B

C

A

Proving Theorem 10.15Paragraph proof: Draw

and . Because C and B intercept the same arc, C B. Likewise, A D. By the AA Similarity Postulate, ∆AEC ∆DEB. So the lengths of corresponding sides are proportional.

ACDB

E

D

B

C

A

ED

EA=EB

EC

EA • EB = EC • ED

Lengths of sides are proportional.

Cross Product Property

Ex. 1: Finding Segment Lengths• Chords ST and PQ

intersect inside the circle. Find the value of x.

6

93

XR

T

S

Q P

RQ • RP = RS • RT Use Theorem 10.15

Substitute values.9 • x = 3 • 6

9x = 18

x = 2

Simplify.

Divide each side by 9.

Using Segments of Tangents and Secants• In the figure shown,

PS is called a tangent segment because it is tangent to the circle at an end point. Similarly, PR is a secant segment and PQ is the external segment of PR.

Q

S

P

R

Theorem 10.16• If two secant

segments share the same endpoint outside a circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment.

C

A

D

E

B

EA • EB = EC • ED

Theorem 10.17• If a secant segment

and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equal the square of the length of the tangent segment.

(EA)2 = EC • ED

C

D

E

A

Ex. 2: Finding Segment Lengths• Find the value of x.

x

10

11

9

S

P

T

R

Q

RP • RQ = RS • RT Use Theorem 10.16

Substitute values.9•(11 + 9)=10•(x + 10)

180 = 10x + 100

80 = 10x

Simplify.

Subtract 100 from each side.

8 = x Divide each side by 10.

Note:

• In Lesson 10.1, you learned how to use the Pythagorean Theorem to estimate the radius of a grain silo. Example 3 shows you another way to estimate the radius of a circular object.

Ex. 3: Estimating the radius of a circle• Aquarium Tank.

You are standing at point C, about 8 feet from a circular aquarium tank. The distance from you to a point of tangency is about 20 feet. Estimate the radius of the tank.

(CB)2 = CE • CD Use Theorem 10.17

Substitute values.

400 16r + 64

336 16r

Simplify.

21 r Divide each side by 16.

(20)2 8 • (2r + 8)

Subtract 64 from each side.

So, the radius of the tank is about 21 feet.

(BA)2 = BC • BD Use Theorem 10.17

Substitute values.

25 = x2 + 4x

0 = x2 + 4x - 25

Simplify.

Use Quadratic Formula.

(5)2 = x • (x + 4)

Write in standard form.

2

)25)(1(444 2

292 x =

Simplify.

Use the positive solution because lengths cannot be negative. So, x = -2 + 3.39.29

x =

Reminders:• Quiz after this section either

Thursday or Friday.

• Test will be after 10.7 next week probably Tuesday or Wednesday.

• Chapter 10 Algebra Review can be done for Extra Credit. Show all work!!!