2018 AMC 12B Problem 4

Below is the professionally curated solution for Problem 4 of the 2018 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2018 AMC 12B solutions, or check the answer key.

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Concepts:chord Pythagorean Theorem circle area

Difficulty rating: 1310

A circle has a chord of length $10,$ and the distance from the center of the circle to the chord is $5.$ What is the area of the circle?

$25\pi$

$50\pi$

$75\pi$

$100\pi$

$125\pi$

Solution:

Dropping a perpendicular from the center to the chord bisects it, forming a right triangle with legs $5$ (half the chord) and $5$ (the distance), and hypotenuse $r.$

Then $r^2=5^2+5^2=50,$ so the area is $\pi r^2=50\pi.$

Thus, the correct answer is B.

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