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This question presents a triangle with known sides and angles, and we are tasked with calculating the length of side BC to three significant figures.

### Given:
- Side \( AB = 5 \, m \)
- Side \( AC = 7 \, m \)
- Angle \( \angle A = 42^\circ \)

We will use the **Cosine Rule** to find the length of side \( BC \). The Cosine Rule is given by:

\[
BC^2 = AB^2 + AC^2 - 2 \cdot AB \cdot AC \cdot \cos(\angle A)
\]

### Substituting the given values:
\[
BC^2 = 5^2 + 7^2 - 2 \cdot 5 \cdot 7 \cdot \cos(42^\circ)
\]
\[
BC^2 = 25 + 49 - 70 \cdot \cos(42^\circ)
\]
\[
BC^2 = 74 - 70 \cdot 0.7431 \quad (\cos(42^\circ) \approx 0.7431)
\]
\[
BC^2 = 74 - 52.017
\]
\[
BC^2 = 21.983
\]
\[
BC = \sqrt{21.983} \approx 4.688 \, m
\]

### Final Answer:
\[
BC \approx 4.69 \, \text{m} \quad (\text{to 3 significant figures})
\]

Would you like further details or explanations?

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Here are 5 related questions to consider:
1. How does the Cosine Rule change if you know all three sides but no angles?
2. Can you use the Sine Rule to find any other sides or angles in this triangle?
3. What is the area of this triangle using the sides and angles provided?
4. How would you solve for side BC if the angle at \( B \) were known instead of \( \angle A \)?
5. How does the accuracy of your trigonometric functions affect your final answer?

**Tip:** When working with trigonometric problems, always ensure your calculator is set to the correct mode (degrees or radians) based on the given angle units.

Work out the length of BC in a triangle where AB = 5m, AC = 7m, and angle A = 42°. Give your answer to 3 significant figures.

Learn how to find the length of side BC in a triangle where AB = 5m, AC = 7m, and angle A = 42°. Using the Cosine Rule, this step-by-step solution provides a detailed explanation suitable for students in Grades 10-12.

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