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choose from which point Then, with the needle of the compass on point N or O or L

The image contains a geometric problem asking to construct a line perpendicular to a given line segment \(NL\) by passing through a specific point. The diagram shows a line \(NL\) and a point \(M\) that is not on the line.

### Task:
To construct a line perpendicular to \(NL\) through the point \(M\) using a compass and straightedge.

### Steps to Solve:
1. **Place the Compass Needle**:
   - Place the compass needle on the point \(M\), the point through which the perpendicular will pass.
   
2. **Construct Arcs**:
   - With \(M\) as the center, draw an arc that intersects the line \(NL\) at two distinct points. These points of intersection will be labeled \(N\) and \(O\).

3. **Draw Perpendicular Line**:
   - Use the two points \(N\) and \(O\) obtained from the previous step.
   - Place the compass needle on \(N\) and draw an arc above and below the line.
   - Without changing the compass width, repeat the same step with the needle on \(O\), creating two intersecting arcs with the ones from \(N\).
   - The intersection points of these arcs will define the line perpendicular to \(NL\) passing through \(M\).

4. **Draw the Perpendicular**:
   - Use a straightedge to draw the line connecting \(M\) and the intersection points from the previous step. This is the required perpendicular line.

**Final Answer**: The needle of the compass should be initially placed on point **M**.

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### Do you want details or have any questions?

Here are some related questions to consider:
1. How can you verify the perpendicularity of the constructed line?
2. What is the significance of choosing specific points for constructing the perpendicular?
3. Can this method be used if the point \(M\) lies on the line \(NL\)?
4. What other geometric constructions can be performed using a compass and straightedge?
5. How would the construction differ if point \(M\) were closer to the line \(NL\)?

**Tip**: When performing geometric constructions, maintaining the compass's radius when drawing arcs from different points ensures the accuracy of the perpendicular line.

To construct a line perpendicular to NL that passes through point M, begin by using a straightedge to draw a line as shown in the diagram. Then, with the needle of the compass on point N or O or L, what should we do?

Learn how to construct a line perpendicular to a given segment NL passing through a point M using a compass and straightedge. This step-by-step guide covers key geometric concepts and methods suitable for students in Grades 6-8.

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