https://file.aiquickdraw.com/mathbot/file/1726075422855evl0tv7y.png

Let's break down the problem from the image step-by-step.

### Given Information:
- Triangle \( LMN \)
- \( LM = 12.4 \, \text{mm} \)
- \( MN = 7.9 \, \text{mm} \)
- \( \angle NLM = 28^\circ \)

### Part (a):
You are asked to calculate the area of triangle \( LMN \), knowing that \( \angle MNL \) is acute.

To calculate the area of a triangle when two sides and the included angle are known, we use the **sine rule for area**:

\[
\text{Area} = \frac{1}{2} \cdot LM \cdot MN \cdot \sin(\angle NLM)
\]

Substituting the given values:
\[
\text{Area} = \frac{1}{2} \cdot 12.4 \, \text{mm} \cdot 7.9 \, \text{mm} \cdot \sin(28^\circ)
\]

Now I will calculate this for you.The area of triangle \( LMN \) is approximately \( 23.0 \, \text{mm}^2 \) to three significant figures.

### Part (b):
If you did not know that \( \angle MNL \) was acute, it could affect your area calculation. This is because the sine function can produce the same value for both acute and obtuse angles (since \( \sin(\theta) = \sin(180^\circ - \theta) \)). If the angle \( MNL \) were obtuse, the true angle would be larger, which would affect the calculation of the triangle's area.

Would you like further details or have any other questions?

Here are five related questions for further practice:
1. How does the sine rule differ from the cosine rule in triangle calculations?
2. What is the difference between acute, obtuse, and right angles?
3. How can you determine whether a triangle has an acute or obtuse angle if only the sides are known?
4. What are some applications of the sine rule in real-life problems?
5. How can you calculate the area of a triangle if only two angles and one side are known?

**Tip**: Always ensure you're using the correct angle when calculating areas involving trigonometry, as this can significantly impact the result.

In triangle LMN, LM = 12.4 mm, MN = 7.9 mm, and angle NLM = 28°. Given that angle MNL is acute, calculate the area of triangle LMN to 3 significant figures.

Learn how to calculate the area of triangle LMN using the sine rule, given sides LM = 12.4 mm, MN = 7.9 mm, and angle NLM = 28°. A detailed step-by-step solution with key concepts in trigonometry and triangle geometry.

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