Example 5 Let the three sides of a right-angled triangle be $a$, $b$, and $c$, all positive integers, and the hypotenuse $c$ satisfies $87 \leqslant c \leqslant 91$. Find the lengths of the three sides of such a right-angled triangle.
【Analysis】According to the representation of Pythagorean triples
$$
\left(m^{2}-n^{2}, 2 m n, m^{2}+n^{2}\right) \text {, }
$$

we can determine the values of $m$ and $n$.
Solution Since Pythagorean triples have the form
$$
\left(m^{2}-n^{2}, 2 m n, m^{2}+n^{2}\right) \text {, }
$$

we set the hypotenuse length as
$$
k\left(m^{2}+n^{2}\right)((m, n)=1, m>n) .
$$
(1) If $k\left(m^{2}+n^{2}\right)=87=3 \times 29$, then when $k=1,29,87$, there is no solution; when $k=3$, $m=5, n=2$, the corresponding Pythagorean triple is $(63,60,87)$.
(2) If $k\left(m^{2}+n^{2}\right)=88=2^{3} \times 11$, checking one by one shows no integer solutions.
(3) If $k\left(m^{2}+n^{2}\right)=89$ is a prime number, then when $k=89$, there is no solution; when $k=1$, $m=8, n=5$, the corresponding Pythagorean triple is $(39,80,89)$.
(4) If $k\left(m^{2}+n^{2}\right)=90$, checking shows that when $k=9$, $m=3, n=1$, the corresponding Pythagorean triple is $(72,54,90)$; when $k=18$, $m=2, n=1$, the corresponding Pythagorean triple is $(54,72,90)$; no other solutions.
(5) If $k\left(m^{2}+n^{2}\right)=91=7 \times 13$, then checking shows that when $k=7$, $m=3, n=2$, the corresponding Pythagorean triple is $(35,84,91)$; no other solutions.
Therefore, the side lengths of such right triangles are $(63,60,87),(39,80,89)$,
$(54,72,90),(35,84,91)$.