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# Circles

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Let line AB and line CD be chords of a circle, that meet at the point Q inside the circle. If AQ = 16, BQ = 12, and CD = 36, then find the minimum length of CQ.

May 14, 2022

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Let x be the minimum length of CQ. Note that by Power of Points formula, AQ * BQ = CQ * DQ.

$$AQ \cdot BQ = CQ \cdot DQ\\ 16 \cdot 12 = x \cdot (36 - x)\\ x^2 - 36x + 192 = 0\\ x = 18 - 2 \sqrt {33}\text{ or }x = 18 + 2\sqrt {33}$$

The smaller one is 18 - 2 * sqrt(33), and that's the answer.

May 15, 2022