1. **State the problem:** We have a trapezoid with the top base labeled as $-4w + 46$, the bottom base as $22$, and the two non-parallel sides marked equal, meaning the trapezoid is isosceles. 2. **Identify what to find:** We need to solve for $w$. 3. **Use the property of isosceles trapezoids:** The legs (non-parallel sides) are equal, and the middle segment labeled $16$ represents the height or the length of the equal legs. 4. **Set up the equation:** Since the trapezoid is isosceles, the difference between the bases is split equally on both sides. The difference between the bases is: $$22 - (-4w + 46) = 22 + 4w - 46 = 4w - 24$$ 5. **Each leg forms a right triangle with half the difference of the bases as one leg and the height $16$ as the other leg.** The length of the leg is $16$, so by the Pythagorean theorem: $$16^2 = \left(\frac{4w - 24}{2}\right)^2 + \text{height}^2$$ But since the legs are equal to $16$, and the height is $16$, the horizontal leg of the triangle is: $$\frac{4w - 24}{2}$$ 6. **Calculate the horizontal leg length:** $$\text{horizontal leg} = \sqrt{16^2 - 16^2} = \sqrt{0} = 0$$ This implies: $$\frac{4w - 24}{2} = 0$$ 7. **Solve for $w$:** $$4w - 24 = 0$$ $$4w = 24$$ $$w = \frac{24}{4} = 6$$ **Final answer:** $$w = 6$$