If the difference between the sides of a right angled triangle is 3 cm and its area is 54 cm²; find its perimeter.

Given:

Area = 54 cm²

Let the two sides of the right-angled triangle be x cm and (x - 3) cm.

Area = $\dfrac{1}{2}$ x base x height

⇒ 54 = $\dfrac{1}{2}$ x BC x AB

⇒ 54 = $\dfrac{1}{2} \times$ (x - 3) $\times$ x

⇒ 54 $\times$ 2 = (x - 3) $\times$ x

⇒ 108 = x² - 3x

⇒ x² - 3x - 108 = 0

⇒ x² - 12x + 9x - 108 = 0

⇒ x(x - 12) + 9(x - 12) = 0

⇒ (x - 12)(x + 9) = 0

⇒ x = 12 or -9

Since length cannot be negative, AB is 12 cm.

BC = (x - 3) cm = 12 - 3 cm = 9 cm

By using the Pythagoras theorem,

AB² + BC² = AC²

⇒ 12² + 9² = AC²

⇒ 144 + 81 = AC²

⇒ 225 = AC²

⇒ AC = $\sqrt{225}$

⇒ AC = 15

Perimeter of right angled triangle = AB + BC + AC

= 12 + 9 + 15 cm

= 36 cm

Hence, the perimeter of the triangle is 36 cm.