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

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#### Solution

Let the base of the right triangle be x cm.

Its altitude = (x - 7) cm

From Pythagoras theorem, we have

Base2 + Altitude2 = Hypotenuse2

∴ x^{2} + (x - 7)^{2} = 132

⇒ 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,

⇒ 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.

Concept: Solutions of Quadratic Equations by Factorization

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