Problem 673

In a right-angled triangle, the hypotenuse is always the longest
side. Hence, the hypotenuse has length $c
$\text{}$ cm}$.

The other two lengths, $a\text{ cm}$
and $b\text{ cm}$, are the lengths of
the legs. Therefore, the area of the triangle is $\dfrac{1}{2}ab\text{ cm}^{2}$.

The area is given to be $\$54\text{ cm}^{2}$$, and so$$\dfrac{1}{2}ab=54$$or$ab=108$.

Since $a$ and $b$ are integers, $a$ and $b$ must form a factor pair of $108$. We are also given that $a<b$, so the only possibilities for the
pair $(a,b)$ are $(1,108)$, $(2,54)$, $(3,36)$, $(4,27)$, $(6,18)$, and $(9,12)$.

Using the Pythagorean Theorem, we must also have that $c=\sqrt{a^2+b^2}$.

Observe that $$\begin{align*} \sqrt{1^2+108^2} &\approx 108.00463 \\ \sqrt{2^2+54^2} &\approx 54.037 \\ \sqrt{3^2+36^2} &\approx 36.1248 \\ \sqrt{4^2+27^2} &\approx 27.2947 \\ \sqrt{6^2+18^2} &\approx 18.9737 \\ \sqrt{9^2+12^2} &= 15\end{align*}$$ and so the only possible
pair $(a,b)$ that leads to an integer
value of $c$ is $a=9$ and $b=12$, so it must be true that $c=15$.