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.