The hypotenuse of a right – angled triangle is 6 cm more than twice the shortest side. If the third side 2 cm less than the hypotenuse, find the sides of the triangle.

Let the shortest side be x cm [Let it be base]


Length of hypotenuse = 2x + 6 [in cm]


Length of other side = Length of hypotenuse – 2 = 2x + 6 – 2 = 2x + 4 [in cm] [Let it be perpendicular]


As we know, By Pythagoras Theorem


(hypotenuse)2 = (base)2 + (perpendicular)2


⇒ (2x + 6)2 = x2 + (2x + 4)2


⇒ 4x2 + 36 + 24x = x2 + 4x2 + 16 + 16x


[(a + b)2 = a2 + b2 + 2ab]


⇒ x2 – 8x – 20 = 0


⇒ x2 – 10x + 2x – 20 = 0


⇒ x(x – 10) + 2(x – 10) = 0


⇒ (x + 2)(x – 10) = 0


⇒ x = – 2 or x = 10 cm


However, Length can't be negative hence x = – 2 is not possible


Therefore,


x = 10 cm


we have,


Shortest Side = x = 10 cm


Hypotenuse = 2x + 6 = 2(10) + 6 = 26 cm


Third side = 2x + 4 = 2(10) + 4 = 24 cm


26