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

Consider a right angle ΔABC

Let the base of the Δ be x

Altitude = x - 7

In the right-angled triangle

According to the Pythagoras theorem

(Hypotenuse)2 = (base)2 + (altitude)2

132 = x2 + (x - 7)2

169 – x2 – x2 - 49 + 14x = 0

2x2 - 14x – 120 = 0

x2 – 7x - 60 = 0

x2 – 12x + 5x - 60 = 0

x( x - 12) + 5( x 12) = 0

(x + 5) (x - 12) = 0

x = - 5 or 12

But base cannot be negative, so x = 12 cm

altitude = 12 - 7 = 5 cm

Hence 5 cm and 12 cm are the two sides of the given triangle.

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