1. **State the problem:** We have an isosceles triangle with two equal sides each of length $\sqrt{74}$, a height (altitude) of length 7, and the base divided into two equal segments each of length $x$. We want to find the value of $x$. 2. **Understand the triangle:** The altitude divides the isosceles triangle into two right triangles. Each right triangle has hypotenuse $\sqrt{74}$, one leg as the altitude 7, and the other leg as $x$ (half the base). 3. **Apply the Pythagorean theorem:** For one right triangle, $$\left(\sqrt{74}\right)^2 = 7^2 + x^2$$ 4. **Simplify the equation:** $$74 = 49 + x^2$$ 5. **Solve for $x^2$:** $$x^2 = 74 - 49 = 25$$ 6. **Find $x$:** $$x = \sqrt{25} = 5$$ **Final answer:** $$x = 5$$