1. **State the problem:** We need to find the length of the hypotenuse $c$ of a right triangle where the legs are 52 mm and 39 mm. 2. **Formula used:** According to the Pythagorean theorem, for a right triangle with legs $a$ and $b$ and hypotenuse $c$, the relationship is: $$c^2 = a^2 + b^2$$ 3. **Apply the formula:** Substitute $a = 52$ mm and $b = 39$ mm: $$c^2 = 52^2 + 39^2$$ 4. **Calculate squares:** $$c^2 = 2704 + 1521$$ 5. **Add the squares:** $$c^2 = 4225$$ 6. **Find the square root:** $$c = \sqrt{4225}$$ 7. **Calculate the square root:** $$c = 65$$ 8. **Conclusion:** The length of the hypotenuse $c$ is 65 millimeters. **Final answer:** $$c = 65 \text{ millimeters}$$