1. **State the problem:** We are given a trapezoid with bases of lengths 4 meters and 12 meters, an area of 48 square meters, and a right side length of 10 meters. We need to find the height $h$ of the trapezoid. 2. **Formula for the area of a trapezoid:** $$\text{Area} = \frac{(\text{base}_1 + \text{base}_2)}{2} \times h$$ where $h$ is the height. 3. **Substitute the known values:** $$48 = \frac{(4 + 12)}{2} \times h$$ 4. **Simplify the bases sum:** $$48 = \frac{16}{2} \times h$$ $$48 = 8h$$ 5. **Solve for $h$ by dividing both sides by 8:** $$h = \frac{48}{8}$$ $$h = 6$$ 6. **Check with the right side length:** The right side is 10 meters, which forms a right triangle with height $h$ and the difference of bases as the base of the triangle. The difference of bases is $12 - 4 = 8$ meters. Using the Pythagorean theorem: $$10^2 = h^2 + 8^2$$ $$100 = h^2 + 64$$ $$h^2 = 36$$ $$h = 6$$ This confirms our height calculation. **Final answer:** $$h = 6 \text{ meters}$$