Lesson 1.3: Basic Constructions

Learning Objectives

• Use a compass and straightedge correctly.
• Construct a segment congruent to a given segment.
• Construct the midpoint of a segment.
• Construct an angle bisector.

Topics Covered

• Compass and Straightedge Rules
• Copying a Segment
• Constructing a Perpendicular Bisector
• Constructing an Angle Bisector

Key Terms

• Construction: A precise geometric drawing using only a compass and straightedge.
• Perpendicular Bisector: A line that divides a segment into two equal parts at \(90^\circ\).
• Angle Bisector: A ray that divides an angle into two congruent angles.
• Congruent: Having the same size and shape.

Lesson Examples

Example 1: Constructing a Segment Bisector

1. Draw segment \( \overline{AB} \).
2. Place the compass at point A and draw arcs above and below the segment.
3. Repeat from point B with the same radius.
4. Draw a line through the intersection points of the arcs.

The line created is the perpendicular bisector of \( \overline{AB} \).

Example 2: Constructing an Angle Bisector

1. Draw angle \( \angle ABC \).
2. Draw an arc that intersects both rays of the angle.
3. From each intersection point, draw arcs that intersect inside the angle.
4. Draw a ray from the vertex to the intersection of the arcs.

Practice Problems

1. Construct a segment congruent to \( \overline{CD} \).
2. Construct the perpendicular bisector of a given segment.
3. Construct an angle bisector and verify the two angles are equal.

Tips & Common Mistakes

• Do not change the compass width when copying distances.
• Use light arcs so the final construction remains clear.
• Remember constructions rely on precision, not estimation.

Summary

Basic constructions allow us to create precise geometric figures using only fundamental tools. Mastery of constructions prepares students for formal geometric proofs and deeper spatial reasoning.

Challenge Problems

1. Construct a perpendicular line through a point not on a given line.
2. Construct a triangle given three side lengths (SSS).
3. Explain why the perpendicular bisector of a segment is the set of all points equidistant from the endpoints.