Basic Constructions

Overview

Geometric constructions involve creating figures or lines using only a compass and straight edge. The basic constructions are fundamental tools for solving complex geometric problems. This note covers constructing angle bisectors, perpendicular bisectors, and perpendicular lines.

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Bisector of a Given Angle

Objective: To divide a given angle into two equal parts.

Method

1. Place the compass at the vertex of the angle and draw an arc that intersects both arms of the angle.
2. Without changing the compass width, place the compass at one intersection point and draw an arc inside the angle.
3. Repeat from the other intersection point.
4. Use a straight edge to draw a line from the vertex to the intersection of the two arcs inside the angle. Result: The line divides the angle into two equal parts.

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Perpendicular Bisector of a Line Segment

Objective: To create a line perpendicular to a given segment that also bisects it into two equal parts.

Method

1. Place the compass at one endpoint of the segment and draw an arc above and below the line. The radius must be more than half the segment length.
2. Repeat the process from the other endpoint, keeping the same compass width.
3. The two arcs intersect above and below the segment.
4. Use a straight edge to draw a line connecting the intersection points of the arcs. Result: The line is the perpendicular bisector of the segment.

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Line Perpendicular to a Given Line Passing Through a Point Not on the Line

Objective: To construct a line perpendicular to a given line that passes through a point not on the line.

Method

1. Place the compass at the given point and draw an arc that intersects the given line at two points.
2. Without changing the compass width, place the compass at each intersection point and draw arcs that intersect above or below the line.
3. Use a straight edge to draw a line connecting the given point to the intersection of the arcs. Result: The line is perpendicular to the given line and passes through the given point.

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Line Perpendicular to a Given Line Passing Through a Point on the Line

Objective: To construct a line perpendicular to a given line that passes through a point on the line.

Method

1. Place the compass at the given point and draw an arc that intersects the line on both sides of the point.
2. Without changing the compass width, place the compass at each intersection point and draw arcs above or below the line.
3. Use a straight edge to draw a line connecting the given point to the intersection of the arcs. Result: The line is perpendicular to the given line and passes through the given point.

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Summary

1. Angle Bisector: Divides an angle into two equal parts.
2. Perpendicular Bisector: Perpendicular line that bisects a segment into two equal parts.
3. Perpendicular from a Point Not on a Line: Creates a perpendicular line from a point outside the line.
4. Perpendicular from a Point on a Line: Creates a perpendicular line through a point on the line. These basic constructions are vital skills for solving more complex geometric problems.