The ratio of the shorter sides of a right-angled triangle is 5 : 12. If the perimeter of the triangle is 360 cm, find the length of the longest side.

Let the length of the shorter side be 5x cm and 12x cm.

Length of hypotenuse = $\sqrt{(5x)^2 + (12x)^2} \text{ cm}$

$= \sqrt{25x^2 + 144x^2} \text{ cm} \\[0.75em] = \sqrt{169x^2} \text{ cm} \\[0.75em] = 13x \text{ cm} \\[0.75em]$

According to given,

$5x + 12x + 13x = 360 \\[0.5em] 30x = 360 \\[0.5em] x = 12.$

∴ x = 12, 13x = 13 $\times$ 12 = 156.

Hence, the length of longest side is 156 cm.