1. **State the problem:** We have a right triangle with legs measuring 5 cm and 17 cm. We need to find the length of the hypotenuse. 2. **Formula used:** The Pythagorean theorem states that in a right triangle, the square of the hypotenuse $c$ is equal to the sum of the squares of the legs $a$ and $b$: $$c^2 = a^2 + b^2$$ 3. **Apply the formula:** Substitute $a = 5$ and $b = 17$: $$c^2 = 5^2 + 17^2$$ 4. **Calculate squares:** $$c^2 = 25 + 289 = 314$$ 5. **Find the hypotenuse:** Take the square root of both sides: $$c = \sqrt{314}$$ 6. **Approximate the value:** $$c \approx 17.7$$ 7. **Answer:** The hypotenuse measures approximately 17.7 cm when rounded to the nearest tenth.