Let AB and CD be chords of a circle, that meet at point Q inside the circle. If AQ = 6, BQ = 15, and CD = 38, then find the minimum length of CQ.
+129907 Intersecting Chord Theorem :
AQ * BQ = CQ * DQ
6 * 15 = CQ * DQ
Let CQ = x and DQ = 38- x....so we have
6 * 15 = x (38 - x)
90 = 38x - x^2 rearrange as
x^2 - 38x + 90 = 0 complete the square on x
x^2 - 38x = -90
x^2 - 38x + 361 = -90 + 361
(x - 19)^2 = 271 take the negative root
x - 19 = -sqrt (271)
x = -sqrt (271) + 19 ≈ 2.54 = minimum length of CQ
