LESSON F: Segment Lengths in Circles

During this lesson, you will find the lengths of segments of chords, tangents and secants in circles.

PART I: FINDING THE LENGTHS OF SEGMENTS 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

THEOREMIf 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.

Name two segments of each chord in the diagram below:

EXAMPLE 1 Finding Segment Lengths

Chords ST and PQ intersect inside the circle. Find the value of x. RQ * XP = SR * RT

9 * x = 3 * 6

9x = 18

x = 2

PART II: USING SEGMENTS OF TANGENTS AND SECANTS

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

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

THEOREM

EXAMPLE 2 Finding Segment Lengths

Find the value of x. RP * RQ = RS * ST

9 * 20 = 10* ( X +10)

180 = 10X + 100

80 = 10X

X = 8

EXAMPLE 3 Estimating the Radius of a CircleYou are standing at point C, about 8 feet from a circular aquarium tank. The distance from you to a point of tangency on the tank is about 20 feet. Estimate the radius of the tank.

The Caribbean Coral Reef Tank at the New England Aquarium is a circular tank 24 feet deep.

Work Space:

(CB)2 = CE * (2r + 8)(20)2 = 8 * (2r + 8)

400 = 16r + 64

336 = 16r

21 ≈ r

Example 4 Finding Segment Lengths

15,: 18: 12 12;15; 50 x + 8; 4

2; 6 9;6 x + 3; x2; 1

Homework Assignment:

Segment Lengths WS