Let the sides at right angles be a and b

And, the third side be c.

Given: a-b = 2 cm

Area of triangle = 24 cm²

Now, since a-b = 2

⇒ a = b + 2

Now we know that,

Area of triangle = 1/2 × base × height

⇒ 24 = 1/2 × b × (b + 2)

⇒ 24 × 2 = b² + 2b

⇒ b² + 2b = 48

⇒ b² + 2b-48 = 0

⇒ b² + 8b-6b-48 = 0

⇒ b(b + 8)-6(b + 8) = 0

⇒ (b + 8)(b-6) = 0

This gives us two equations,

i. b + 8 = 0

⇒ b = -8

ii. b-6 = 0

⇒ b = 6

Since, the side of the triangle cannot be negative

Therefore, b = 6 cm

⇒ a = (b + 2) cm

⇒ a = (6 + 2) cm

⇒ a = 8 cm

Now we know that,

Base² + Perpendicular² = Hypotenuse²

⇒ a² + b² = c²

⇒ 8² + 6² = c²

⇒ c² = 64 + 36

⇒ c² = 100

⇒ c = 10

Now,

Perimeter of triangle = a + b + c

⇒ Perimeter of triangle = 8 + 6 + 10

⇒ Perimeter of triangle = 24 cm