**Solution:**

Let x cm be the base of the right triangle.

Given, the altitude of right triangle = (x – 7) cm

From Pythagoras’ theorem, we know,

Base^{2} + Altitude^{2} = Hypotenuse^{2}

∴ x^{2 }+ (x – 7)^{2} = 13^{2}

⇒ 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

Thus, either x – 12 = 0 or x + 5 = 0,

⇒ x = 12 or x = – 5

Since sides cannot be negative, 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.