AMC 10 Weekly Practice Round 3

Complete problem set with solutions and individual problem pages

Problem 21 Medium

A point P is chosen at random inside a circle C with radius 2. What is the probability that the chord with midpoint P has length less than 2\sqrt{3}?

  • A.

    \frac{1}{2}

  • B.

    \frac{3}{4}

  • C.

    \frac{1}{4}

  • D.

    \frac{3}{10}

  • E.

    \frac{3}{5}

Answer:B

From the problem we have:

 

The chord with midpoint P has length less than 2\sqrt{3} if and only if the distance from the circle’s center to P satisfies

d > \sqrt{2^{2} - \left(\tfrac{2\sqrt{3}}{2}\right)^{2}} = 1.

 

That is, when |CP| > 1, the chord with midpoint P has length less than 2\sqrt{3}.

 

By the geometric probability formula, the required probability is

P = \frac{\pi \cdot 2^{2} - \pi \cdot 1^{2}}{\pi \cdot 2^{2}} = \tfrac{3}{4}.

 

Therefore, the answer is \tfrac{3}{4}.

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