1. **Stating the problem:** We are given a quadrilateral ABCD with diagonals AC and DB intersecting at right angles. The lengths of AC and DB are given as 10 and 17 respectively, and we need to find the area $A$ of the quadrilateral. 2. **Formula used:** For a quadrilateral with perpendicular diagonals, the area $A$ is given by: $$A = \frac{1}{2} \times AC \times DB$$ This formula applies because the diagonals intersect at right angles, effectively dividing the quadrilateral into four right triangles. 3. **Substitute the given values:** $$A = \frac{1}{2} \times 10 \times 17$$ 4. **Calculate the area:** $$A = \frac{1}{2} \times 170 = 85$$ 5. **Interpretation:** The area of the quadrilateral ABCD is 85 square units. 1. **Second problem:** Given AC = 16 and DB = 30, find the area $A$. 2. **Apply the same formula:** $$A = \frac{1}{2} \times AC \times DB$$ 3. **Substitute values:** $$A = \frac{1}{2} \times 16 \times 30$$ 4. **Calculate:** $$A = \frac{1}{2} \times 480 = 240$$ 5. **Interpretation:** The area of the quadrilateral is 240 square units. **Final answers:** - For AC = 10 and DB = 17, $A = 85$ - For AC = 16 and DB = 30, $A = 240$