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. X P S Q R 4 3 x 8 6 8 x 3 = 6 x 4

Example Another way to solve would be to think of two numbers that multiply to 36 and add to 15. b. If AP = 4, PB = 9, and CD = 15, find CP. P B A D C x 4 P 9 15 - x 9 x 4 = x(15 – x) 36 = 15x – x2 x2 – 15x + 36 = 0 (x )(x ) = 0 - - 3 12 CP is either 12 or 3. x – 3 = 0 or x – 12 = 0 If it were given that the figure was drawn to scale, then CP = 12. x = 3 or x = 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. G H F D E FD x FE = FH x FG External x Whole Thing = External x Whole Thing

Example a. If PQ = 9, QR = 3, and TR = 4, then SR = ____. b. If PQ = 11, SR = 12, and TR = 5, find PR. R Q T S P 9 3(12) = 4x 12 36 = 4x x PR = x + 11 = (4) + 11 = 15 9 = x 3 4 RS = 9 x(x + 11) = 5(12) R Q T S P x2 + 11x = 60 x2 + 11x – 60 = 0 11 x + 11 (x )(x ) = 0 - + 4 15 12 x – 4 = 0 or x + 15 = 0 x 5 x = 4 or x = -15 x must be positive

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. B P C A PB x PC = (PA)2 External x Whole Thing = (Tangent)2

Example 25 5 a. If RT = 25 and RS = 5, find QR. b. IF QR = 6 and ST = 9, find RS. x x2 = 5(25) x2 = 125 x = 5√5 QR = 5√5 x + 9 9 R Q S T x x(x + 9) = 62 x2 + 9x = 36 x must be positive. x2 + 9x – 36 = 0 6 (x )(x ) = 0 - + 3 12 x – 3 = 0 or x + 12 = 0 x = 3 or x = -12 RS = 3

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

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

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

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