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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.

Feb 8, 2021

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Let x be the length of CQ. Then DQ = 30 - x.

By power of points,

x(30 - x) = 72

x^2 - 30x + 72 = 0

(x - 15)^2 - 225 + 72 = 0

(x - 15)^2 = 153

$$x = 15\pm \sqrt{153}$$

Minimum length of CQ = $$15 - \sqrt{153}$$

Feb 8, 2021