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2018 AMC 12B Problem 4

Problem 4 of 25EasierGeometry

A circle has a chord of length 10,10, and the distance from the center of the circle to the chord is 5.5. What is the area of the circle?

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Solution

Dropping a perpendicular from the center to the chord bisects it, forming a right triangle with legs 55 (half the chord) and 55 (the distance), and hypotenuse r.r. Then r2=52+52=50,r^2=5^2+5^2=50, so the area is πr2=50π.\pi r^2=50\pi. Thus, the correct answer is B.

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Concepts: chord · Pythagorean Theorem · circle area

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.