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The image presents a geometry problem about triangle \( LMN \). Here's the transcription of the problem:

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In triangle \( LMN \):

- \( LM = 12.4 \, \text{mm} \)
- \( MN = 7.6 \, \text{mm} \)
- \( \angle NLM = 28^\circ \)

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### Questions:

**a)** Given that angle \( MNL \) is acute, calculate the area of triangle \( LMN \). Give your answer to 3 significant figures.

**b)** Write a sentence to explain what effect not knowing that angle \( MNL \) was acute would have on your calculation for the area of \( LMN \).

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### To solve part (a):

We will use the **formula for the area of a triangle** when two sides and the included angle are known:
\[
\text{Area} = \frac{1}{2}ab \sin C
\]
Where:
- \( a = LM = 12.4 \, \text{mm} \)
- \( b = MN = 7.6 \, \text{mm} \)
- \( C = \angle NLM = 28^\circ \)

Let me compute the area using these values.

The area of triangle \( LMN \) is approximately \( 22.1 \, \text{mm}^2 \), rounded to three significant figures.

### Part (b):

If it was not known that angle \( MNL \) is acute, there would be uncertainty about whether to use the acute or obtuse version of the angle when calculating the area. This could result in an incorrect area calculation because the sine function produces different results for acute and obtuse angles (since \( \sin(\theta) \) for an obtuse angle is positive but smaller than for an acute angle).

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Do you have any questions or want more details? Here are five related questions to further your understanding:

1. How would you calculate the area of a triangle if all three sides are given?
2. How does the sine rule help in solving triangle problems like this?
3. What is the difference between acute, right, and obtuse angles in a triangle?
4. How would the calculation change if the angle given was obtuse?
5. Can the cosine rule be used here instead? Why or why not?

**Tip:** Always remember to check if your angle is acute or obtuse when using trigonometric formulas for area to avoid errors.

In triangle LMN:
- LM = 12.4 mm
- MN = 7.6 mm
- Angle NLM = 28°
a) Given that angle MNL is acute, calculate the area of triangle LMN.
b) Explain the effect of not knowing that angle MNL was acute on the area calculation.

Learn how to calculate the area of triangle LMN using the sine rule, with LM = 12.4 mm, MN = 7.6 mm, and angle NLM = 28°. This problem explains the use of trigonometry to solve for areas of triangles with acute angles, and discusses how an incorrect assumption about the angle type can affect the result.

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