1. **Problem Statement:** We are given a right triangle with one leg of length $x$, another leg of length 15, and the hypotenuse of length 17. We need to find the value of $x$. 2. **Formula Used:** According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse ($c$) is equal to the sum of the squares of the other two sides ($a$ and $b$): $$c^2 = a^2 + b^2$$ 3. **Assigning Values:** Here, the hypotenuse $c = 17$, one leg $b = 15$, and the other leg $a = x$ (unknown). 4. **Apply the Pythagorean theorem:** $$17^2 = x^2 + 15^2$$ 5. **Calculate squares:** $$289 = x^2 + 225$$ 6. **Isolate $x^2$:** $$x^2 = 289 - 225$$ $$x^2 = 64$$ 7. **Solve for $x$:** $$x = \sqrt{64}$$ $$x = 8$$ 8. **Conclusion:** The value of $x$ is 8. This means the vertical side length of the triangle is 8 units.