Perpendicular Line Construction

This section focuses on constructing a perpendicular line through a specific point on a given line.

Steps to draw a perpendicular to a point P on line AB:

1. Place compass at P and draw an arc intersecting AB at another point C
2. With compass at A, draw an arc above AB
3. Repeat from C with the same compass width
4. Draw a line from P through the intersection of the upper arcs

Highlight: This method ensures the constructed line is exactly perpendicular to the original line at the chosen point.

The lesson includes practice problems for students to apply these construction techniques:

1. Construct the perpendicular bisector of a given line segment
2. Construct the angle bisector of a given angle
3. Construct the perpendicular bisector of a line segment labeled CD

Example: These problems provide opportunities to practice precise compass and ruler use while reinforcing the concepts of perpendicular and angle bisectors.

Constructions 1: Perpendicular and Angle Bisectors

This lesson introduces fundamental geometric construction techniques using a compass and ruler. Students will learn how to construct perpendicular bisectors of lines and angle bisectors.

Definition: A perpendicular bisector cuts a line in half at right angles.

The lesson begins with a review of key terms:

Vocabulary:

• Perpendicular: Lines that intersect at a right (90 degree) angle
• Intersect: To cross over or have a common point
• Arc: Any smooth curve joining two points

A step-by-step worked example demonstrates how to construct a perpendicular bisector of a line:

1. Draw a line of given length
2. Open compass to more than half the line's length
3. Draw arcs from each endpoint
4. Connect intersection points of arcs

Highlight: Construction marks should not be erased, as they are part of the solution.

Constructing Perpendicular and Angle Bisectors

This page provides detailed instructions for constructing perpendicular and angle bisectors using a compass and ruler.

To construct the perpendicular bisector of a line AB:

1. Place compass at A, set over halfway and draw two arcs
2. Repeat from B with the same compass width
3. Draw the perpendicular bisector through the arc intersection points

To construct the angle bisector of angle BAC:

1. Place compass at A and draw an arc crossing AB and AC
2. From the intersections, draw two arcs that intersect
3. Draw the angle bisector from A through the new intersection point

Example: These techniques can be applied to bisect angles of various measures, such as 60°, 90°, or 110°.

The page also covers how to construct a perpendicular from a point not on the line:

1. Draw an arc from the point P intersecting the line at two points
2. Use these intersection points as centers to draw intersecting arcs below the line
3. Connect P to the new intersection point to create the perpendicular line