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1959 AMC 12 Problem 31

Attempt Problem 31 of the 1959 AMC 12 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 1959 AMC 12 solutions, or check the answer key.

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

31.

A square, with an area of 40,40, is inscribed in a semicircle. The area of a square that could be inscribed in the entire circle with the same radius is:

8080

100100

120120

160160

200200

Answer: B
Concepts:circlesquare (geometry)Pythagorean Theorem
Difficulty rating: 1360
Small Hint:

Let the first square have side ss and place its lower side on the semicircle’s diameter

Big Hint:

A top vertex has horizontal coordinate s2\frac{s}{2} and vertical coordinate ss relative to the center

Solution:

Let the semicircle have radius rr and the square have side s,s, where s2=40.s^2=40. A top vertex of the square is s2\frac{s}{2} horizontally and ss vertically from the center, so r2=s2+(s2)2=54s2=50. r^2=s^2+\left(\frac s2\right)^2=\frac54s^2=50. A square inscribed in the full circle has diagonal 2r,2r, hence area 2r2=100.2r^2=100.

Thus, the correct answer is B.

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