1. **State the problem:** We have a right triangle with legs $a$ and $b$, and hypotenuse $c$. Given $c=13$ and $a=12$, find the length of side $b$. 2. **Formula used:** According to the Pythagorean theorem for right triangles, $$c^2 = a^2 + b^2$$ This means the square of the hypotenuse equals the sum of the squares of the legs. 3. **Substitute known values:** $$13^2 = 12^2 + b^2$$ 4. **Calculate squares:** $$169 = 144 + b^2$$ 5. **Isolate $b^2$:** $$b^2 = 169 - 144$$ 6. **Simplify:** $$b^2 = 25$$ 7. **Find $b$ by taking the square root:** $$b = \sqrt{25}$$ 8. **Evaluate the square root:** $$b = 5$$ **Final answer:** $$\boxed{5}$$