2007 Cyprus MO/Lyceum/Problem 7

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Problem

If a diagonal $d$ of a rectangle forms a $60^\circ$ angle with one of its sides, then the area of the rectangle is

$\mathrm{(A) \ } \frac{d^2 \sqrt{3}}{4}\qquad \mathrm{(B) \ } \frac{d^2}{2}\qquad \mathrm{(C) \ } 2d^2\qquad \mathrm{(D) \ } d^2 \sqrt{2}\qquad \mathrm{(E) \ } \mathrm{None\;of\;these}$

Solution

Using $30-60-90$ right triangle ratios, the lengths of the sides of the rectangle are $\frac{d}{2}$ and $\frac{d\sqrt{3}}{2}$.

The area of the rectangle is $\frac{d^2\sqrt{3}}{4}\Longrightarrow\mathrm{ A}$.

See also

2007 Cyprus MO, Lyceum (Problems)
Preceded by
Problem 6
Followed by
Problem 8
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