Answer
$7$ ft and $24$ ft
Work Step by Step
Let $x$ and $y$ be the lengths of the other two sides of the right triangle.
We have
$\frac{1}{2}xy=84$
$x^2+y^2=25^2$
\Equivalently,
$y=\frac{168}{x}$
$x^2+y^2-625=0$
Substituting the first equation to the second one, we have
$x^2+\frac{168^2}{x^2}-625=0$ $(\times x^2$)
$x^4-625x^2+168^2=0$
$x^4-(576+49)x^2+576\cdot 49=0$
$(x^2-576)(x^2-49)=0$
$x^2=576$ or $x^2=49$
$x=24$ or $x=7$
If $x=24$, then $y=\frac{168}{24}=7$.
If $x=7$, then $y=\frac{168}{7}=24$.
So, the lengths of the other two sides of the triangle are $7$ ft and $24$ ft.
