1950 AHSME Problems/Problem 9

Problem

The area of the largest triangle that can be inscribed in a semi-circle whose radius is $r$ is:

$\textbf{(A)}\ r^{2}\qquad\textbf{(B)}\ r^{3}\qquad\textbf{(C)}\ 2r^{2}\qquad\textbf{(D)}\ 2r^{3}\qquad\textbf{(E)}\ \frac{1}{2}r^{2}$

Solution

The area of a triangle is $\frac12 bh.$ To maximize the base, let it be equal to the diameter of the semi circle, which is equal to $2r.$ To maximize the height, or altitude, choose the point directly in the middle of the arc connecting the endpoints of the diameter. It is equal to $r.$ Therefore the area is $\frac12 \cdot 2r \cdot r = \boxed{\mathrm{(A) }r^2.}$

1950 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
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1950 AHSME Problems/Problem 9

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