1. **State the problem:** We have a right triangle where the altitude is 7 cm less than the base, and the hypotenuse is 13 cm. We need to find the lengths of the other two sides (base and altitude). 2. **Define variables:** Let the base be $x$ cm. Then the altitude is $x - 7$ cm. 3. **Apply the Pythagorean theorem:** For a right triangle with legs $a$, $b$ and hypotenuse $c$, we have: $$a^2 + b^2 = c^2$$ Here, $a = x$, $b = x - 7$, and $c = 13$. So, $$x^2 + (x - 7)^2 = 13^2$$ 4. **Expand and simplify:** $$x^2 + (x^2 - 14x + 49) = 169$$ $$2x^2 - 14x + 49 = 169$$ 5. **Bring all terms to one side:** $$2x^2 - 14x + 49 - 169 = 0$$ $$2x^2 - 14x - 120 = 0$$ 6. **Simplify by dividing all terms by 2:** $$x^2 - 7x - 60 = 0$$ 7. **Factor the quadratic:** $$x^2 - 7x - 60 = (x - 12)(x + 5) = 0$$ 8. **Solve for $x$:** $$x - 12 = 0 \Rightarrow x = 12$$ $$x + 5 = 0 \Rightarrow x = -5$$ (not possible since length cannot be negative) 9. **Find the altitude:** $$x - 7 = 12 - 7 = 5$$ 10. **Final answer:** The base is 12 cm and the altitude is 5 cm.