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Two chords TY and OP of a circle intersect at point K such that TK = 2, KY = 16, and KP = 2*(KO). Find the length OP.

1. 8
2. 10
3. 12
4. 14

Correct answer: 12

Solution

By the intersecting chords theorem, TK*KY = OK*KP. Let KO = a, KP = 2a. Then 2*16 = a*2a => 32 = 2a² => a² = 16 => a = 4. So KO = 4, KP = 8, and OP = KO + KP = 4 + 8 = 12.

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