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The problem involves a triangle \( FGH \) with the following given dataThe problem involves a triangle \( FGH \) with the following given 
- \( FG = 10.2 \, \text{mm} \)
- \( GH = 7.3 \, \text{mm} \)
- \( \angle HFG = 38^\circ \)

### Part a) 
We need to calculate the area of triangle \( FGH \), assuming that angle \( \angle GHF \) is acute. The formula for the area of a triangle when two sides and the included angle are known is:
\[
\text{Area} = \frac{1}{2} \cdot a \cdot b \cdot \sin(C)
\]
where \( a \) and \( b \) are the sides and \( C \) is the included angle.

In this case:
- \( a = FG = 10.2 \, \text{mm} \)
- \( b = GH = 7.3 \, \text{mm} \)
- \( \angle HFG = 38^\circ \)

Thus, the area can be calculated as:
\[
\text{Area} = \frac{1}{2} \cdot 10.2 \, \text{mm} \cdot 7.3 \, \text{mm} \cdot \sin(38^\circ)
\]
Let me calculate this for you.The area of triangle \( FGH \) is approximately \( 22.921 \, \text{mm}^2 \) to three significant figures.

### Part b) 
If it were not known that angle \( GHF \) is acute, the area calculation could be affected because the assumption that the given angle \( \angle HFG = 38^\circ \) accurately represents the orientation of the triangle would be unclear. If \( GHF \) were not acute, the triangle might have a different configuration (for instance, \( GHF \) could be obtuse), which would affect the actual height or base used in the area calculation, leading to an incorrect result.

Would you like further details or have any other questions? Here are five related questions for you:

1. How does the sine function relate to calculating triangle areas?
2. What happens to the area of a triangle if one of its angles exceeds \(90^\circ\)?
3. How would you calculate the height of triangle \( FGH \) if the base were known?
4. What other methods can be used to find the area of a triangle?
5. How does knowing the type of angle (acute or obtuse) impact geometric calculations?

**Tip:** When using trigonometric formulas for area, ensure you are using the correct units for angles (degrees vs. radians).

In triangle FGH, FG = 10.2 mm, GH = 7.3 mm, and angle HFG = 38°. a) Given that angle GHF is acute, calculate the area of triangle FGH. Give your answer to 3 significant figures. b) Write a sentence to explain what effect not knowing that GHF was acute would have on your calculation for the area of FGH.

This problem involves calculating the area of triangle FGH using trigonometric methods. Given side lengths FG = 10.2 mm, GH = 7.3 mm, and angle HFG = 38°, learn how to apply the sine rule to find the area. Explore how the assumption that angle GHF is acute affects the calculation. Suitable for high school students in Grades 10-12.

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