5. A rectangle has perimeter 10 and diagonal $\sqrt{15}$. What is its area?
Solution:
5
If the sides are $x$ and $y$, we have $2 x+2 y=10$, so $x+y=5$, and $\sqrt{x^{2}+y^{2}}=\sqrt{15}$, so $x^{2}+y^{2}=15$. Squaring the first equation gives $x^{2}+2 x y+y^{2}=25$, and subtracting the second equation gives $2 x y=10$, so the area is $x y=5$.