If the sum of two smaller sides of a right-angled triangle is 17 cm and the perimeter is 30 cm, then find the area of the triangle.

Let one of the two smaller sides be x cm, then the other side is (17 - x) cm.

Length of hypotenuse = perimeter - sum of other two sides = 30cm - 17cm = 13cm

In right angled triangle

Perpendicular² + Base² = Hypotenuse²

∴ x² + (17 - x)² = 13²

$\Rightarrow x^2 + 289 + x^2 - 34x = 169 \\[1em] \Rightarrow 2x^2 - 34x + 289 - 169 = 0 \\[1em] \Rightarrow 2x^2 - 34x + 120 = 0 \\[1em] \Rightarrow 2(x^2 - 17x + 60) = 0 \\[1em] \Rightarrow x^2 - 17x + 60 = 0 \\[1em] \Rightarrow x^2 - 12x - 5x + 60 = 0 \\[1em] \Rightarrow x(x - 12) - 5(x - 12) = 0 \\[1em] \Rightarrow (x - 5)(x - 12) = 0 \\[1em] \Rightarrow x - 5 = 0 \text{ or } x - 12 = 0 \\[1em] \Rightarrow x = 5 \text{ or } x = 12$

If x = 5 , 17 - x = 12 and if x = 12 , 17 - x = 5.

Hence, two small sides are 5 , 12.

Area of right angled triangle = $\dfrac{1}{2} \times \text{base} \times \text{height}$

∴ $\dfrac{1}{2} \times 5\text{cm} \times 12\text{cm} = 30 cm^2$

Hence the area of triangle is 30cm².