1. **Problem statement:** We have a rectangle between two right triangles. The entire length of the figure is 31, the hypotenuse of each triangle is 19, and the left side of the rectangle is 17. We need to find $x$, the top side length of the rectangle. 2. **Understanding the problem:** The figure consists of two right triangles and a rectangle in the middle. The total length (base) is 31, and the height of the rectangle (left side) is 17. The hypotenuse of each triangle is 19. 3. **Using the Pythagorean theorem:** For each right triangle, if we let the base be $b$ and the height be $h$, then: $$19^2 = b^2 + h^2$$ 4. **Assigning variables:** The height of the rectangle is 17, so the height of the triangles is also 17 (since the rectangle is between them). Thus, $h = 17$. 5. **Calculate the base of each triangle:** $$19^2 = b^2 + 17^2$$ $$361 = b^2 + 289$$ $$b^2 = 361 - 289 = 72$$ $$b = \sqrt{72} = 6\sqrt{2}$$ 6. **Calculate the total base length of the two triangles:** $$2b = 2 \times 6\sqrt{2} = 12\sqrt{2}$$ 7. **Find $x$, the top side of the rectangle:** The total length is 31, so: $$x = 31 - 2b = 31 - 12\sqrt{2}$$ **Final answer:** $$x = 31 - 12\sqrt{2}$$