1956 AHSME Problems/Problem 37

On a map whose scale is $400$ miles to an inch and a half, a certain estate is represented by a rhombus having a $60^{\circ}$ angle. The diagonal opposite $60^{\circ}$ is $\frac {3}{16}$ in. The area of the estate in square miles is:

$\textbf{(A)}\ \frac{2500}{\sqrt{3}}\qquad \textbf{(B)}\ \frac{1250}{\sqrt{3}}\qquad \textbf{(C)}\ 1250\qquad \textbf{(D)}\ \frac{5625\sqrt{3}}{2}\qquad \textbf{(E)}\ 1250\sqrt{3}$

Solution

When we construct the diagram, we can see that the rhombus is made up of two equilateral triangles. The short diagonal in the rhombus is one side of both equilateral triangles, so we can simply scale the length up and find the area that way.

We know that the scale is $400$ miles to $\frac{3}{2}$ inches, so we simply divide by $8$ to find that the short diagonal of the estate is $50$ miles.

When plugged into the formula, we get $2*50^2\frac{\sqrt{3}}{4}$ (we have two congruent triangles to work with), which converts to $\boxed{\textbf{(E) }1250\sqrt{3}}$ square miles.

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1956 AHSME Problems/Problem 37

Solution

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