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# Circle

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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

### 1+0 Answers

#1
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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