Q5.The altitude of right triangle is 7 cm less than its base. If, hypotenuse is 13 cm. Find the other two sides.

Let the base of the right triangle be \(x\).

The altitude can be represented by \(x - 7\).

In a right triangle,

\(Hypotenuse^2 = Base^2 + Altitude^2\) (Pythagoras Theorem)

\(13^2 = x^2 + (x - 7)^2 \)

\(169 = x^2 + x^2 - 14x + 49 \)

\(2x^2 - 14x - 120 = 0 \)

\(x^2 - 7x - 60 = 0\)

(Splitting * -7x * as * 5x - 12x *)

\(x^2 + 5x - 12x - 60 = 0\)

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

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

The roots of this equation are the values of x for which \( (x - 12)(x + 5) = 0 \)

which are,

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

Thus, \(x = -5\) or \(x = 12\)

Since the length of any side cannot be negative, we reject \(x = -5\).

**Thus, the length of the base is 12 cm and the length of the altitude is 12 - 7 = 5 cm. **