1. **Problem statement:** We have a quadrilateral with equal diagonals. The segments joining the midpoints of opposite sides have lengths 13 and 7. We need to find the area of this quadrilateral. 2. **Key formula:** For any quadrilateral, the segment joining the midpoints of opposite sides relates to the diagonals. If the diagonals are equal, the area $A$ can be found using the formula: $$A = \frac{1}{2} \times d \times m$$ where $d$ is the length of the diagonals (equal) and $m$ is the length of the segment joining midpoints of opposite sides. 3. **Important property:** The segments joining midpoints of opposite sides in a quadrilateral are half the length of the diagonals. Since there are two such segments, their lengths correspond to half the diagonals projected in different directions. 4. **Given:** The two segments joining midpoints of opposite sides are 13 and 7. These correspond to half the lengths of the diagonals projected along two directions. Since diagonals are equal, the length of each diagonal $d$ satisfies: $$2 \times 13 = d \quad \text{and} \quad 2 \times 7 = d$$ But this is contradictory unless these segments correspond to different components. 5. **Using Varignon's theorem:** The quadrilateral formed by joining midpoints of the sides is a parallelogram whose sides are half the diagonals of the original quadrilateral. 6. **Length of diagonals:** The segments joining midpoints of opposite sides are the sides of this parallelogram. So the parallelogram has sides 13 and 7, which are half the diagonals of the original quadrilateral. 7. **Therefore, the diagonals of the original quadrilateral are:** $$d_1 = 2 \times 13 = 26$$ $$d_2 = 2 \times 7 = 14$$ 8. **Area formula for quadrilateral with diagonals $d_1$, $d_2$ and angle $\theta$ between them:** $$A = \frac{1}{2} d_1 d_2 \sin \theta$$ 9. **Since diagonals are equal, $d_1 = d_2 = d$, but here $d_1 \neq d_2$. So the problem states diagonals are equal, but given segments imply different diagonal lengths.** 10. **Assuming the problem means the diagonals are equal in length, then the segments joining midpoints of opposite sides are equal. But given lengths are 13 and 7, so the problem likely means the segments joining midpoints of opposite sides are equal to 13 and 7 respectively, which correspond to half the diagonals.** 11. **Hence, the diagonals are:** $$d = 2 \times 13 = 26$$ $$d = 2 \times 7 = 14$$ 12. **Since diagonals are equal, take the average:** $$d = \frac{26 + 14}{2} = 20$$ 13. **Area of quadrilateral with equal diagonals $d$ and angle $\theta$ between them:** $$A = \frac{1}{2} d^2 \sin \theta$$ 14. **Using the parallelogram formed by midpoints, the lengths 13 and 7 are sides of the parallelogram, so the area of this parallelogram is:** $$A_p = 13 \times 7 \times \sin \theta$$ 15. **Area of original quadrilateral is twice the area of this parallelogram:** $$A = 2 A_p = 2 \times 13 \times 7 \times \sin \theta = 182 \sin \theta$$ 16. **Since diagonals are equal, the parallelogram is a rhombus, so $\sin \theta = 1$ (max area).** 17. **Therefore, the area of the quadrilateral is:** $$A = 182$$ **Final answer:** The area of the quadrilateral is 182 square units.