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)

\(\Rightarrow 13^2 = x^2 + (x - 7)^2 \)
\(\Rightarrow 169 = x^2 + x^2 - 14x + 49 \)
\(\Rightarrow 2x^2 - 14x - 120 = 0 \)
\(\Rightarrow x^2 - 7x - 60 = 0\)
(Splitting -7x as 5x - 12x )
\(\Rightarrow x^2 + 5x - 12x - 60 = 0\)
\(\Rightarrow x(x + 5) - 12(x + 5) = 0\)
\(\Rightarrow (x - 12)(x + 5) = 0\)

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.