The difference of lengths of sides forming right angle in right angled triangle is 3 cm. If the perimeter of the triangle is 36 cm. Find the area of the triangle.

Let the length of perpendicular be x.

Now, given that the difference of perpendicular and base = 3

⇒ Base = x + 3

Also, Perimeter = 36

⇒ 36 = x + (x + 3) + Hypotenuse

⇒ Hypotenuse = 36 – x – x – 3

⇒ 33 – 2x

Now, using Pythagoras Theorem,

Hypotenuse² = Base² + Perpendicular²

⇒ (33 – 2x)² = x² + (x + 3)²

⇒ 1089 + 4x² – 132x = x² + x² + 9 + 6x

⇒ 2x² – 1138x + 1080 = 0

Dividing whole equation by 2, we get –

⇒ x² – 69x + 540 = 0

⇒ x² – (60 + 9)x + 540 = 0

⇒ x² – 60x – 9x + 540 = 0

⇒ x(x – 60) – 9(x – 60) = 0

⇒ (x – 60)(x – 9) = 0

⇒ x = 60 or x = 9

∵ Perimeter is given to be 36 cm

⇒ x(Perpendicular) = 9 cm

⇒ Base = x + 3

= 12 cm

Now,

⇒ 54 cm²