Segment Lengths in Circles Geometry Mrs Padilla Spring

Objectives/Assignment • Find the lengths of segments of chords. • Find the lengths of

Finding the Lengths of Chords • When two chords intersect in the interior of

Theorem 10. 15 • If two chords intersect in the interior of a circle,

Proving Theorem 10. 15 • You can use similar triangles to prove Theorem 10.

Proving Theorem 10. 15 Paragraph proof: Draw and. Because C and B intercept the

Ex. 1: Finding Segment Lengths • Chords ST and PQ intersect inside the circle.

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

Theorem 10. 16 • If two secant segments share the same endpoint outside a

Theorem 10. 17 • If a secant segment and a tangent segment share an

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

Note: • In Lesson 10. 1, you learned how to use the Pythagorean Theorem

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

(CB)2 = CE • CD (20)2 8 • (2 r + 8) 400 16

(BA)2 = BC • BD (5)2 = x • (x + 4) Use Theorem

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Segment Lengths in Circles Geometry Mrs. Padilla Spring 2012

Objectives/Assignment • Find the lengths of segments of chords. • Find the lengths of segments of tangents and secants.

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. 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. = 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. RQ • RP = RS • RT Use Theorem 10. 15 9 • x=3 • 6 Substitute values. Simplify. 9 x = 18 x=2 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.

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 EA • EB = EC • ED external segment equals the product of the length of the other secant segment and the length of its external segment.

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 (EA)2 = EC • ED and the length of its external segment equal the square of the length of the tangent segment.

Ex. 2: Finding Segment Lengths • Find the value of x. RP • RQ = RS • RT Use Theorem 10. 16 9 • (11 + 9)=10 • (x + 10) Substitute values. Simplify. 180 = 10 x + 100 Subtract 100 from each side. 80 = 10 x 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.

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 (20)2 8 • (2 r + 8) 400 16 r + 64 336 16 r 21 r Use Theorem 10. 17 Substitute values. Simplify. Subtract 64 from each side. Divide each side by 16. So, the radius of the tank is about 21 feet.

(BA)2 = BC • BD (5)2 = x • (x + 4) Use Theorem 10. 17 25 = x 2 + 4 x 0 = x 2 + 4 x - 25 Substitute values. Simplify. Write in standard form. x= Use Quadratic Formula. x= Simplify. Use the positive solution because lengths cannot be negative. So, x = -2 + 3. 39.