2018 AMC 12B Problem 4

Attempt Problem 4 of the 2018 AMC 12B below, then check your answer against the professionally curated solution 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.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

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

The radius, half the chord, and the distance to the chord form a right triangle

With legs

5

and

5,

the radius satisfies

r^2=5^2+5^2

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.

Problem 4 in other years