Answer
7 ft and 24 ft.
Work Step by Step
Step 1. Assume the lengths of the other two sides are $x$ ft and $y$ ft.
Step 2. Use the Pythagorean's Theorem, we have $x^2+y^2=25^2=625$
Step 3. The area of the triangle can be represented as $\frac{1}{2}xy=84$
Step 4. Thus, we have the following system of equations:
\begin{cases} x^2+y^2=625 \\ xy=168 \end{cases}
Step 5. The second equation gives $y=\frac{168}{x}$, use it to substitute the variable in the first equation, we have:
$x^2+(\frac{168}{x})^2=625$
Step 6. Multiply $x^2$ on both sides, we have $x^4-625x^2+168^2=0$
Step 7. We can solve the above equation by letting $z=x^2$ and use a quadratic formula, however, we can just graph the function as shown and find the solutions as $x=7, 24$ ft.
Step 8. Back-substitute $x$-values into the second equation to get $y=24, 7$ft.
Step 9. We conclude the lengths of the other sides as 7 ft and 24 ft.
