1. **State the problem:** We have an isosceles triangle with two equal sides of length 10 and a height (altitude) of 8 drawn from the top vertex to the base. We need to find the length of the base $x$. 2. **Understand the triangle:** The altitude divides the isosceles triangle into two right triangles, each with hypotenuse 10, one leg 8 (the altitude), and the other leg half of the base, $\frac{x}{2}$. 3. **Apply the Pythagorean theorem:** For one right triangle, $$10^2 = 8^2 + \left(\frac{x}{2}\right)^2$$ 4. **Calculate:** $$100 = 64 + \frac{x^2}{4}$$ 5. **Isolate $x^2$:** $$100 - 64 = \frac{x^2}{4}$$ $$36 = \frac{x^2}{4}$$ 6. **Multiply both sides by 4:** $$144 = x^2$$ 7. **Take the square root:** $$x = \sqrt{144}$$ 8. **Simplify:** $$x = 12$$ **Final answer:** The length of the base $x$ is 12.