10.5 Segment Lengths in Circles

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# 10.5 Segment Lengths in Circles - PowerPoint PPT Presentation

10.5 Segment Lengths in Circles. Geometry Mrs. 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.

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### 10.5 Segment Lengths in Circles

Geometry

Mrs. 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.
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.Theorem 10.15

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

=

Proving Theorem 10.15

Paragraph 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.

Lengths of sides are proportional.

EA • EB = EC • ED

Cross Product Property

RQ • RP =RS •RT

Use Theorem 10.15

9 • x = 3 • 6

Substitute values.

Simplify.

9x = 18

Divide each side by 9.

x = 2

Ex. 1: Finding Segment Lengths
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.Using Segments of Tangents and Secants
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.Theorem 10.16

EA• EB = EC • ED

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.Theorem 10.17

(EA)2 = EC • ED

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

RP • RQ = RS • RT

Use Theorem 10.16

9•(11 + 9)=10•(x + 10)

Substitute values.

Simplify.

180 = 10x + 100

Subtract 100 from each side.

80 = 10x

Divide each side by 10.

8 = x

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.
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.Ex. 3: Estimating the radius of a circle

Solution

(CB)2 = CE • CD

Use Theorem 10.17

(20)28 • (2r + 8)

Substitute values.

Simplify.

400 16r + 64

Subtract 64 from each side.

336  16r

Divide each side by 16.

21  r

Ex. 4: Finding

Segment Lengths

(BA)2 = BC • BD

Use Theorem 10.17

(5)2 =x • (x + 4)

Substitute values.

Simplify.

25 = x2 + 4x

Write in standard form.

0 = x2 + 4x - 25