1. **Problem statement:** We have a trapezoid with diagonals of lengths 6 and 10, and the segment connecting the midpoints of the bases (the mid-segment) is 4. We want to find the lengths of the bases. 2. **Relevant formula:** In a trapezoid, the segment connecting the midpoints of the bases (mid-segment) is equal to half the sum of the lengths of the two bases. That is, $$\text{mid-segment} = \frac{b_1 + b_2}{2}$$ where $b_1$ and $b_2$ are the lengths of the two bases. 3. **Given:** mid-segment $= 4$, diagonals $= 6$ and $10$. 4. From the mid-segment formula: $$4 = \frac{b_1 + b_2}{2} \implies b_1 + b_2 = 8$$ 5. **Using the trapezoid diagonal property:** The sum of the squares of the diagonals equals the sum of the squares of the two bases plus twice the square of the mid-segment: $$d_1^2 + d_2^2 = b_1^2 + b_2^2 + 2 \times (\text{mid-segment})^2$$ 6. Substitute the known values: $$6^2 + 10^2 = b_1^2 + b_2^2 + 2 \times 4^2$$ $$36 + 100 = b_1^2 + b_2^2 + 2 \times 16$$ $$136 = b_1^2 + b_2^2 + 32$$ 7. Simplify: $$b_1^2 + b_2^2 = 136 - 32 = 104$$ 8. Use the identity: $$(b_1 + b_2)^2 = b_1^2 + 2 b_1 b_2 + b_2^2$$ 9. Substitute $b_1 + b_2 = 8$ and $b_1^2 + b_2^2 = 104$: $$8^2 = 104 + 2 b_1 b_2$$ $$64 = 104 + 2 b_1 b_2$$ 10. Solve for $b_1 b_2$: $$2 b_1 b_2 = 64 - 104 = -40$$ $$b_1 b_2 = -20$$ 11. Since the product of the bases is negative, this is impossible for lengths (which must be positive). This suggests an inconsistency or error in the problem data. **Conclusion:** Given the diagonals 6 and 10 and mid-segment 4, the bases cannot be real positive lengths. Please verify the problem data. **If you want to find the bases assuming the data is consistent, the sum of the bases is 8, but the product is negative, which is impossible for lengths. Hence, no valid base lengths exist under these conditions.