A rectangular parking lot has a diagonal of $25$ meters and an area of $168$ square meters. In meters, what is the perimeter of the parking lot? $\textbf{(A)}\ 52 \qquad$ $\textbf{(B)}\ 58 \qquad$ $\textbf{(C)}\ 62 \qquad$ $\textbf{(D)}\ 68 \qquad$ $\textbf{(E)}\ 70$
Solution
Let the side lengths of the rectangular parking lot be $a$ and $b$ meters. Since its diagonal is $25$ meters, the Pythagorean theorem gives $$a^2+b^2=25^2=625.$$ Its area is $168$ square meters, so $ab=168$.
Use $(a+b)^2=a^2+b^2+2ab$ to find the square of the sum of its side lengths: $$(a+b)^2=625+2(168)=961=31^2.$$ Since $a$ and $b$ are positive, $a+b=31$.
The perimeter is $2(a+b)=2(31)=62$ meters, so the answer is $\textbf{(C)}$.
