1951 AHSME Problems/Problem 32

Problem

If $\triangle ABC$ is inscribed in a semicircle whose diameter is $AB$ , then $AC+BC$ must be

$\textbf{(A)}\ \text{equal to }AB\qquad\textbf{(B)}\ \text{equal to }AB\sqrt{2}\qquad\textbf{(C)}\ \geq AB\sqrt{2}\qquad\textbf{(D)}\ \leq AB\sqrt{2}$ $\textbf{(E)}\ AB^{2}$

Solution

Because $AB$ is the diameter of the semi-circle, it follows that $\angle C = 90$ . Now we can try to eliminate all the solutions except for one by giving counterexamples.

$\textbf{(A):}$ Set point $C$ anywhere on the perimeter of the semicircle except on $AB$ . By triangle inequality, $AC+BC>AB$ , so $\textbf{(A)}$ is wrong. $\textbf{(B):}$ Set point $C$ on the perimeter of the semicircle infinitesimally close to $AB$ , and so $AC+BC$ almost equals $AB$ , therefore $\textbf{(B)}$ is wrong. $\textbf{(C):}$ Because we proved that $AC+BC$ can be very close to $AB$ in case $\textbf{(B)}$ , it follows that $\textbf{(C)}$ is wrong. $\textbf{(E):}$ Because we proved that $AC+BC$ can be very close to $AB$ in case $\textbf{(B)}$ , it follows that $\textbf{(E)}$ is wrong. Therefore, the only possible case is $\boxed{\textbf{(D)}\ \leq AB\sqrt{2}}$

1951 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 31	Followed by Problem 33
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1951 AHSME Problems/Problem 32

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