https://file.aiquickdraw.com/mathbot/file/1726710647586006knbpk.jpg

The problem in the image asks for using a ruler and a compass to solve a triangle, ΔLMN, where:

- \( \angle LMN = 45^\circ \)
- \( LM = 7 \, \text{cm} \)
- \( MN = 8 \, \text{cm} \)

Additionally, the question (b) asks to measure and state the length of \( LN \).

### Steps to solve the problem:

1. **Draw the base \( LM \):**
   - Use a ruler to draw a line segment \( LM \) with a length of 7 cm.

2. **Use the angle \( \angle LMN = 45^\circ \):**
   - Use a protractor to measure a \( 45^\circ \) angle at point \( M \).

3. **Construct the segment \( MN \):**
   - From point \( M \), use a compass set to 8 cm to draw an arc. The point where this arc intersects the line extending from \( M \) (forming a \( 45^\circ \) angle) is point \( N \).

4. **Measure the length of \( LN \):**
   - After constructing the triangle, use the ruler to measure the distance from point \( L \) to point \( N \), which gives you the length of \( LN \).

Would you like to discuss any further details about this construction or need clarification?

### Follow-up Questions:
1. How would you verify that the angle \( \angle LMN \) is exactly \( 45^\circ \)?
2. What type of triangle is formed if \( \angle LMN = 45^\circ \)?
3. Can you use the Pythagorean theorem in this case? Why or why not?
4. How does the length of \( LN \) relate to the other sides?
5. If \( \angle LMN \) were a different angle, how would the length of \( LN \) change?

**Tip:** When using a compass and ruler, accuracy in measurement and drawing is essential to ensure precise results!

Using ruler and compass, construct ΔLMN in which ∠LMN = 45°, LM = 7 cm, MN = 8 cm. Measure and state the length of LN.

Learn how to construct a triangle ΔLMN where ∠LMN = 45°, LM = 7 cm, and MN = 8 cm using a ruler and compass. Includes step-by-step guidance for measuring and calculating the length of LN. Suitable for students in Grades 6-8.

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