5. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

Solution: Let us say, the base of the right triangle be x cm. Given, the altitude of right triangle = (x – 7) cm From Pythagoras theorem, we know, Base2 + Altitude2 = Hypotenuse2 ∴ x2 + (x – 7)2 = 132 ⇒ x2 + x2 + 49 – 14x = 169 ⇒ 2x2 – 14x – 120 = 0 ⇒ x2 – 7x – 60 = 0 ⇒ x2 – 12x + 5x – 60 = 0 ⇒ x(x – 12) + 5(x – 12) = 0 ⇒ (x – 12)(x + 5) = 0 Thus, either x – 12 = 0 or x + 5 = 0, ⇒ x = 12 or x = – 5 Since sides cannot be negative, x can only be 12. Therefore, the base of the given triangle is 12 cm and the altitude of this triangle will be (12 – 7) cm = 5 cm.