AMC 12 2015a Problem 8

The ratio of the length to the width of a rectangle is $4$ : $3$. If the rectangle has diagonal of length $d$, then the area may be expressed as $kd^2$ for some constant $k$. What is $k$?

\textbf{(A)}\ \frac27 \textbf{(B)}\ \frac37 \textbf{(C)}\ \frac{12}{25} \textbf{(D)}\ \frac{16}{25} \textbf{(E)}\ \frac34