1. **Problem Statement:** We have a triangle LMN with LM = 12.4 mm, MN = 7.9 mm and angle NLM = 28^6. We want to calculate the area of the triangle given that angle MNL is acute and then discuss the impact of not knowing this. 2. **Known Data:** - LM = 12.4 mm - MN = 7.9 mm - \angle NLM = 28^6 (angle at L) 3. **Approach:** Use the sine area formula for a triangle: $$\text{Area} = \frac{1}{2}ab\sin C$$ where $a$ and $b$ are two sides surrounding angle $C$. 4. **Identifying sides and angle:** The given angle is $28^6$ at vertex L, between sides LM and LN. However, LN is unknown. Since only LM and MN are given, and $\angle NLM$ is given, we can use the Law of Cosines or use the sine rule to find the missing side or the included angle opposite a side to use the formula. 5. **Step:** Use Law of Cosines to find side LN: $$LN^2 = LM^2 + MN^2 - 2\times LM \times MN \times \cos(\angle MNL)$$ But $\angle MNL$ is unknown. Since the problem focuses on $\angle MNL$ being acute, we assume it's acute but unknown. 6. Since the triangle has LM and MN and angle NLM, area can be computed directly using: $$\text{Area} = \frac{1}{2} \times LM \times MN \times \sin(\angle NLM)$$ 7. Substitute values: $$\text{Area} = \frac{1}{2} \times 12.4 \times 7.9 \times \sin 28^6$$ Calculate $\sin 28^6 \approx 0.4695$. 8. Calculate area: $$\text{Area} = 0.5 \times 12.4 \times 7.9 \times 0.4695 \\ = 0.5 \times 12.4 \times 7.9 \times 0.4695 \\ = 0.5 \times 12.4 \times 3.705 \approx 0.5 \times 45.922 = 22.961$$ 9. Round the answer to 3 significant figures: $$\text{Area} = 23.0 \text{ mm}^2$$ 10. **Answer for a):** The area of triangle LMN is approximately 23.0 mm\(^2\). 11. **Answer for b):** Not knowing that $\angle MNL$ is acute would create ambiguity in the triangle's shape. Specifically, if $\angle MNL$ were obtuse, the height of the triangle would alter, potentially changing the sine of the included angle and thus the area. The area calculation would be incorrect or ambiguous without the acute angle assumption, as the sine of an obtuse angle differs and can lead to a different area. Thus, knowing that $\angle MNL$ is acute ensures a unique, correct area calculation.