The difference between the two perpendicular sides of a right-angled triangle is 2 cm and its area is 24 cm². What is the perimeter (in cm) of the triangle?

Given:

Difference = 2 cm and Area = 24 cm².

Formula Used:

Area of right-angled triangle = (1/2) × base × height

Pythagorean theorem: a² + b² = c²

Calculation:

Let the lengths of the perpendicular sides be (x + 2) cm and x cm.

Area = (1/2) × (x + 2) × x = 24

⇒ (x + 2) × x = 48

⇒ x² + 2x - 48 = 0

Solving the quadratic equation x² + 2x - 48 = 0:

⇒ (x + 8) (x - 6) = 0

⇒ x = 6 (since length cannot be negative)

So, the sides are 6 cm and 8 cm.

Hypotenuse, c = √ (6² + 8²)

⇒ c = √ (36 + 64)

⇒ c = √100

⇒ c = 10 cm

Perimeter = 6 + 8 + 10

⇒ Perimeter = 24 cm

∴ The correct answer is option (4).