1. **Problem statement:** We have a pentagon composed of a rectangle and an isosceles triangle on top. The rectangle has height $27$ and the triangle has two equal sides of length $14$. We want to find the length of the base of the triangle (which is also the top side of the rectangle). 2. **Known values:** - Height of rectangle $= 27$ - Equal sides of triangle $= 14$ 3. **Goal:** Find the base length of the triangle. 4. **Approach:** The triangle is isosceles with two equal sides $14$ and an unknown base $b$. The altitude from the apex to the base splits the base into two equal segments of length $\frac{b}{2}$ and forms two right triangles. 5. **Using the Pythagorean theorem:** Let the altitude of the triangle be $h$. Then: $$14^2 = h^2 + \left(\frac{b}{2}\right)^2$$ 6. **Altitude $h$ is the vertical height above the rectangle, but since the rectangle height is $27$, the total height from the bottom to the apex is $27 + h$. However, the problem only gives the rectangle height and the equal sides of the triangle, so we focus on the triangle alone.** 7. **We need more information to find $b$ or $h$. Since the problem does not provide the altitude or the total height, we assume the altitude is the height of the triangle above the rectangle.** 8. **If the altitude is given or can be found, we can solve for $b$. Otherwise, the problem is incomplete.** Since the problem does not provide the altitude or base length, we cannot find a numeric answer for $b$ without additional information. **Final note:** To solve for the base $b$ of the isosceles triangle with equal sides $14$ and altitude $h$, use: $$b = 2\sqrt{14^2 - h^2}$$ Without $h$, the base length cannot be determined.