The perimeter of a right-angled triangle is five times the length of itsshortest side. The numerical value of the area of the triangle is 15 times thenumerical value of the length of the shortest side. Find the lengths of thethree sides of the triangle.

⇒ a + b + (a ² +b ² ) ^1/2 = 5a … (i)

⇒ 1/2ab = 15a … (ii)

⇒ ab = 30a

∴ b = 30

From (i),
a + b +  (a ² +b ² ) ^1/2 = 5a
(a ² +b ² ) ^1/2 = 5a – a – b
= 4a – b
Squaring both sides,
a ² + b ² = (4a – b) ²
⇒ a ² + b ² = 16a ² + b ² – 8ab
⇒ 15a ² =8ab

∴ a = 16

∴ The sides of the triangle are 16 cm, 30 cm and 34 cm.