1. **State the problem:** We need to find the value of $x$ in a right triangle where the sides are 17, 20, and $x$, with the right angle between the sides 17 and $x$. 2. **Formula used:** According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. The formula is: $$c^2 = a^2 + b^2$$ where $c$ is the hypotenuse, and $a$ and $b$ are the legs. 3. **Identify the hypotenuse:** The longest side is 20, so $c = 20$, and the legs are 17 and $x$. 4. **Apply the formula:** $$20^2 = 17^2 + x^2$$ 5. **Calculate squares:** $$400 = 289 + x^2$$ 6. **Isolate $x^2$:** $$x^2 = 400 - 289 = 111$$ 7. **Find $x$ by taking the square root:** $$x = \sqrt{111} \approx 10.5357$$ 8. **Round to the nearest tenth:** $$x \approx 10.5$$ **Final answer:** $x \approx 10.5$