1. **State the problem:** We have a right triangle with two sides given: one side is 13 units, and the other perpendicular side is 12 units. We need to find the length of the third side. 2. **Identify the sides:** In a right triangle, the longest side opposite the right angle is the hypotenuse. Here, 13 is the hypotenuse because it is the longest side. 3. **Use the Pythagorean theorem:** The theorem states that for a right triangle with legs $a$ and $b$, and hypotenuse $c$, the relationship is: $$c^2 = a^2 + b^2$$ 4. **Assign values:** Let the unknown side be $x$. Since $c=13$ and one leg is 12, we have: $$13^2 = 12^2 + x^2$$ 5. **Calculate squares:** $$169 = 144 + x^2$$ 6. **Isolate $x^2$:** $$x^2 = 169 - 144$$ $$x^2 = 25$$ 7. **Find $x$ by taking the square root:** $$x = \sqrt{25}$$ $$x = 5$$ 8. **Conclusion:** The length of the third side is 5 units. This is exact, so no rounding is necessary.