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