Let AB and CD be chords of a circle, that meet at point Q inside the circle. If AQ = 6, BQ = 12, and CD = 30, then find the minimum length of CQ.

Guest May 7, 2022

#1**0 **

By the intersecting chord theorem, we have this equation: \(AQ \times BQ = CQ \times DQ\)

Substituting what we know, we have: \(6 \times 12 = DQ \times CQ\)

Let \(CQ = x\) and \(DQ = y\).

We can form the system: \(xy = 72\) and \(x+y = 30\)

Now, we have to solve for and take the smaller option for x.

Can you do it from here?

Hint: You will substitute into a quadratic, and you will have to use the formula: \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

BuilderBoi May 7, 2022