Geometry 9 7 Circles and Lengths of Segments

Geometry 9. 7 Circles and Lengths of Segments

Theorem • When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord. Q S 4 8 8 x 3=6 x 4 P x. X 3 6 R

Example b. If AP = 4, PB = 9, and CD = 15, find CP. Another way to solve would be to think of two numbers that multiply to 36 and add to 15. A 4 B 15 - x 9 x 4 = x(15 – x) 36 = 15 x – x 2 D x P 9 C x 2 – 15 x + 36 = 0 (x - 3 )(x - 12 ) = 0 x – 3 = 0 or x – 12 = 0 x = 3 or x = 12 CP is either 12 or 3. If it were given that the figure was drawn to scale, then CP = 12.

Theorem • When two secant segments are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment. FD x FE = FH x FG E D G H External x Whole Thing = External x Whole Thing F

Example a. If PQ = 9, QR = 3, and TR = 4, then SR = ____. b. If PQ = 11, SR = 12, and TR = 5, find PR. P S 3(12) = 4 x 36 = 4 x 9=x RS = 9 9 12 PR = x + 11 = (4) + 11 = 15 x(x + 11) = 5(12) R x 2 + 11 x = 60 (x - 4 )(x + 15) = 0 x – 4 = 0 or x + 15 = 0 x = 4 or x = -15 4 3 P x 2 + 11 x – 60 = 0 T Q S 11 x + 11 T Q 5 x R x must be positive 12 x

Theorem • When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segment is equal to the square of the tangent segment. A PB x PC = (PA)2 P C External x Whole Thing = (Tangent)2 B

Example 25 T a. If RT = 25 and RS = 5, find QR. b. IF QR = 6 and ST = 9, find RS. S 5 R x x 2 = 5(25) x 2 = 125 x = 5√ 5 Q x+9 QR = 5√ 5 x(x + 9) = 62 x 2 + 9 x = 36 x 2 + 9 x – 36 = 0 x R (x - 3 )(x + 12) = 0 S x must be positive. 6 Q x – 3 = 0 or x + 12 = 0 x = 3 or x = -12 9 RS = 3 T

Complete. 3 1. If RQ=8, QS = 6, and TQ = 16, then QU = ____. 14 2. If RS =13, RQ = 7, and QU = 3, then TQ = ____. 12 3. If TU = 10√ 2, QU = 2√ 2 , and RQ = 8, then RS ____. 8 or 9 4. If RS = 17, TQ = 18, and QU = 4, then RQ = ___. Who can fill in the picture on the front board? R U S Q Q T

Complete. 8½ 5. If DF = 17, EF = 3, and GF = 6, then HF = ____. 5 6. If HG = 8, GF = 7, and DF = 21, then EF = ____. 6 7. If GF = 8, HG = 10, and DE = 18, then EF = ____. 5√ 2 8. If DF = 25, EF = 4, and HG = GF = x, then x = ____. D E F G H

Secants and tangents are shown. Find the value of x. 9) 10) 11) 12) 15 4 x 5 x x=6 12 x x=3 16 x=9 8 3 x x=4 x

HW • Be nice to someone, take the trash out • P. 363 -366 WE(1 -9, 13 -20, 24)