1. **Stating the problem:** We need to find the area of a parallelogram given its diagonals and one side length. 2. **Formula used:** The area of a parallelogram can be found using the formula involving diagonals and the angle between them: $$\text{Area} = \frac{1}{2} \times d_1 \times d_2 \times \sin(\theta)$$ where $d_1$ and $d_2$ are the lengths of the diagonals and $\theta$ is the angle between the diagonals. 3. **Important rules:** The diagonals of a parallelogram bisect each other but are not necessarily perpendicular. Without the angle between diagonals, we cannot directly use the diagonal formula unless the parallelogram is a rhombus or rectangle. 4. **Given data:** Diagonals $d_1 = 80$ cm, $d_2 = 60$ cm, and a side length $s = 30$ cm. 5. **Approach:** Since the side length is given, we can use the formula for the area of a parallelogram: $$\text{Area} = \text{base} \times \text{height}$$ But height is not given. Alternatively, use the formula for the area in terms of sides and the sine of the angle between them: $$\text{Area} = ab \sin(\alpha)$$ where $a$ and $b$ are adjacent sides and $\alpha$ is the angle between them. 6. **Finding the angle between sides:** We can use the law of cosines on the diagonals: $$d_1^2 + d_2^2 = 2(a^2 + b^2)$$ But we only have one side $s=30$ cm, so assume the parallelogram has sides $a=30$ cm and $b$ unknown. 7. **Using the formula for diagonals in parallelogram:** $$d_1^2 + d_2^2 = 2(a^2 + b^2)$$ Substitute known values: $$80^2 + 60^2 = 2(30^2 + b^2)$$ $$6400 + 3600 = 2(900 + b^2)$$ $$10000 = 2(900 + b^2)$$ $$\cancel{2}5000 = \cancel{2}(900 + b^2)$$ $$5000 = 900 + b^2$$ $$b^2 = 5000 - 900 = 4100$$ $$b = \sqrt{4100} \approx 64.03 \text{ cm}$$ 8. **Finding the angle between sides:** Use the law of cosines on one diagonal: $$d_1^2 = a^2 + b^2 - 2ab \cos(\theta)$$ $$80^2 = 30^2 + 64.03^2 - 2 \times 30 \times 64.03 \cos(\theta)$$ $$6400 = 900 + 4100 - 3841.8 \cos(\theta)$$ $$6400 = 5000 - 3841.8 \cos(\theta)$$ $$6400 - 5000 = -3841.8 \cos(\theta)$$ $$1400 = -3841.8 \cos(\theta)$$ $$\cos(\theta) = -\frac{1400}{3841.8} \approx -0.3645$$ 9. **Calculate $\sin(\theta)$:** $$\sin(\theta) = \sqrt{1 - \cos^2(\theta)} = \sqrt{1 - (-0.3645)^2} = \sqrt{1 - 0.1328} = \sqrt{0.8672} \approx 0.9317$$ 10. **Calculate area:** $$\text{Area} = ab \sin(\theta) = 30 \times 64.03 \times 0.9317 \approx 1790.9 \text{ cm}^2$$ 11. **Compare with options:** None of the options exactly match 1790.9 cm², but the closest is 2400 cm² (option C). **Conclusion:** The answer 2400 cm² is not exactly correct based on calculations; the actual area is approximately 1791 cm².