1. **Problem statement:** We have a right triangle ABC with a right angle at B. Given: - AB = $x$ (vertical side) - BC = 8 (horizontal side) - AC = 10 (hypotenuse) We need to find the value of $x$. 2. **Formula used:** In a right triangle, the Pythagorean theorem applies: $$AC^2 = AB^2 + BC^2$$ This means the square of the hypotenuse equals the sum of the squares of the other two sides. 3. **Apply the formula:** Substitute the known values: $$10^2 = x^2 + 8^2$$ Simplify the squares: $$100 = x^2 + 64$$ 4. **Isolate $x^2$:** $$x^2 = 100 - 64$$ $$x^2 = 36$$ 5. **Solve for $x$:** Take the square root of both sides: $$x = \pm \sqrt{36}$$ $$x = \pm 6$$ 6. **Interpret the result:** Since $x$ represents a length, it must be positive: $$x = 6$$ **Final answer:** $x = 6$ This means the vertical side AB is 6 units long.