Question

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

Answer 1

Let the base of the right triangle be x cm.
Its altitude = (x − 7) cm

from Pythagoras theorem,
$\text{Base}^2 + \text{Altitude} ^2 = \text{Hypoteneous}^2$
∴ $x^2 + (x+7) ^2 = 13^2$
⇒ $x^2 + x^2 +49-14x = 169$
⇒ $2x^2 -14x -120 =0$
⇒ $x^2 -7x -60=0$
⇒ $x^2 -12x+5x -60=0$
⇒ $x(x-12)+5(x-12)=0$
⇒ $(x-12)(x+5)=0$

Either $x − 12 = 0$ or $x + 5 = 0$,
i.e., $x = 12$ or $x = −5$

Since sides are positive, 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.