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

Given,

Hypotenuse = 13 cm

Let base = x cm, so perpendicular = (x - 7) cm.

By pythagoras theorem,

⇒ Hypotenuse² = Perpendicular² + Base²

⇒ 13² = (x - 7)² + x²

⇒ 169 = x² + 49 - 14x + x²

⇒ 169 = 2x² - 14x + 49

⇒ 2x² - 14x + 49 - 169 = 0

⇒ 2x² - 14x - 120 = 0

⇒ 2(x² - 7x - 60) = 0

⇒ x² - 7x - 60 = 0

⇒ x² - 12x + 5x - 60 = 0

⇒ x(x - 12) + 5(x - 12) = 0

⇒ (x + 5)(x - 12) = 0

⇒ x + 5 = 0 or x - 12 = 0

⇒ x = -5 or x = 12.

Since, side cannot be negative.

∴ Base(x) = 12 cm, Altitude (x - 7) = 5 cm.

Hence, other two sides are 5 cm and 12 cm.