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Level 1

In the diagram, point Y Y is on the circle and point P P lies outside the circle such that P Y PY is tangent to the circle. A A is a point on the circle such that segment P A PA meets the circle again at point B . B.

If P A = 22 PA = 22 and P Y = 10 , PY = 10, then what is A B ? AB?

Express your answer as a decimal rounded to the nearest hundredth.


The answer is 17.45.

Jonathan Hsu by Jonathan Hsu , 19, USA

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1 solution

Alex Zhong
Apr 8, 2015

According to the Power of a Point Theorem: P B = P Y 2 P A . PB=\dfrac{PY^2}{PA}.

Therefore, A B = P A − P B = P A 2 − P Y 2 P A = 2 2 2 − 1 0 2 22 = 192 11 = 17.4545 . AB = PA- PB = \dfrac{PA^2-PY^2}{PA} = \dfrac{22^2-10^2}{22}=\dfrac{192}{11}=\boxed{17.4545}.

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