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.

 May 7, 2022

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: \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

 May 7, 2022

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