1. **Stating the problem:** We need to find the area of a kite-shaped quadrilateral with diagonals intersecting at right angles. The kite has sides 13, 13, 15, and 15, and one diagonal length is given as 24. 2. **Formula used:** The area of a kite is given by $$\text{Area} = \frac{d_1 \times d_2}{2}$$ where $d_1$ and $d_2$ are the lengths of the diagonals. 3. **Important rule:** The diagonals of a kite are perpendicular, so they intersect at right angles. 4. **Finding the unknown diagonal:** Let the diagonals be $d_1 = 24$ (given) and $d_2$ (unknown). The kite can be split into four right triangles by the diagonals. 5. **Using the Pythagorean theorem:** The diagonals intersect at right angles and bisect each other into segments. Let the half-lengths of the diagonals be $12$ and $x$ respectively. 6. **Using the sides:** The two pairs of adjacent sides are 13 and 15. Using the right triangles formed, we have $$13^2 = 12^2 + x^2$$ $$169 = 144 + x^2$$ $$x^2 = 25$$ $$x = 5$$ 7. **Full length of the other diagonal:** Since $x$ is half of $d_2$, $$d_2 = 2 \times 5 = 10$$ 8. **Calculate the area:** $$\text{Area} = \frac{24 \times 10}{2} = 12 \times 10 = 120$$ **Final answer:** The area of the kite is $120$ square units.