Let $\overline{AB}$ and $\overline{CD}$ be chords of a circle, that meet at point $Q$ inside the circle. If $AQ = 6,$ $BQ = 12$, and $CD = 38$, then find the minimum length of $CQ$.
TYSM!!!!
+128474 Intersecting Chord Theorem
Let CQ = x and DQ = 38 - x
We have that
AQ * BQ = CQ * DQ
6 * 12 = x ( 38 - x)
72 = 38x - x^2 rearrange as
x^2 - 38x + 72 = 0 factor as
(x - 36) ( x - 2) = 0
Setting each factor to 0 and solving for x we have
x = 36 reject
x = 2 = min length of CQ
