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

Let the base = x cm Therefore the height = x – 7 cm
Given that : hypotenuse = 13 cm
Using Pythagoras theorem, x² + (x – 7)² = 13²
⇒ x² + x² – 14x + 49 = 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
⇒ (x -12) = 0 or (x + 5) = 0
Either x = 12 or x = – 5
But x ≠ – 5, as x is side of triangle.
Therefore, x = 12 and the second side = x – 7 = 12 – 7 = 5
Hence, the other two sides are 5 cm and 12 cm.