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Difference between revisions of "1959 AHSME Problems/Problem 10"

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

Revision as of 12:58, 16 July 2024

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 $\frac{1}{2}AB * AC *\sin\angle BAC = \frac{1}{2}AD * AE *\sin\angle DAE$. Since $\angle BAC = \angle DAE$, we have $AB*AC = AD*AE$, so that $3.6*3.6 = 1.2*AE$. Therefore, $AE = \frac{3.6^2}{1.2} = 10.8$. Thusly, our answer is $\boxed{\text{(D)}}$, and we are done.

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