1959 AHSME Problems/Problem 10

Problem

In $\triangle ABC$ with $\overline{AB}=\overline{AC}=3.6$ , a point $D$ is taken on $AB$ at a distance $1.2$ from $A$ . Point $D$ is joined to $E$ in the prolongation of $AC$ so that $\triangle AED$ is equal in area to $ABC$ . Then $\overline{AE}$ is: $\textbf{(A)}\ 4.8 \qquad\textbf{(B)}\ 5.4\qquad\textbf{(C)}\ 7.2\qquad\textbf{(D)}\ 10.8\qquad\textbf{(E)}\ 12.6$

Solution

Note that $[\triangle ABC]=[\triangle ADE]$ , so $\frac{1}{2}AB * AC *\sin\angle BAC = \frac{1}{2}AD * AE *\sin\angle DAE$ . Since $\angle BAC = \angle DAE$ , we have $AB*AC = AD*AE$ , so that $3.6*3.6 = 1.2*AE$ . Therefore, $AE = \frac{3.6^2}{1.2}=\boxed{\textbf{(D) }10.8}$ .

1959 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
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1959 AHSME Problems/Problem 10

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