1950 AMC 12 Problem 9

Attempt Problem 9 of the 1950 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 1950 AMC 12 solutions, or check the answer key.

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

The area of the largest triangle that can be inscribed in a semicircle whose radius is

r

is:

r^2

r^3

2r^2

2r^3

\dfrac12r^2

The longest possible base is the diameter of the semicircle

With the diameter as base, the greatest possible altitude is the radius

A triangle in the semicircle has base at most the diameter

2r

and altitude at most

r.

Therefore its area is at most

\frac12(2r)(r)=r^2.

This bound is attained by using the diameter as the base and the topmost point of the semicircle as the third vertex.

Thus, the correct answer is A.

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