Construction involves working with angles and lines to scale without a protractor. Constructions are performed using a straight edge and a compass. There are four main constructions that you need to be able to perform.

Equipment

A straight edge is a ruler without any markings that indicate length.

A compass is an instrument that consists of a pointed tip and a hole to place a pencil. It is used to draw circles by rotating the compass about the tip.

Perpendicular bisector of a line

Definition

Two lines are perpendicular if they meet at right angles (or $90\degree$).

A line is bisected if it is split exactly in half.

The perpendicular bisector of a line is another line that splits the line in half while meeting it at right angles.

The grey line above is a perpendicular bisector of the line $RS$.

Constructing the perpendicular bisector of a line

procedure

1.

Set the compass to a length that is greater than half the length of the line.

2.

Using this set length, draw arcs about both ends of the line.

3.

Using a straight edge, draw a line between the two points where the arcs intersect. This line is the perpendicular bisector.

Or, presented visually:

Note: When performing constructions, always leave the construction lines visible. Do not rub them out!

Angle bisector

Definition

The angle bisector of an angle is a line that splits an angle exactly in half.

Constructing the angle bisector of an angle

procedure

1.

Set the compass to a set length. Draw an arc about the point where the angle is so that the arc touches the two lines that the angle is between.

2.

At each of these points where the arc meets the lines, draw an arc with the same set length. This will result in two more arcs: one from each point.

3.

Draw a straight line from the vertex to the point where both arcs meet. This line is the angle bisector.

Or, presented visually:

Perpendicular to a line through a point not on the line

This construction involves drawing a line through a given point that is perpendicular to another line.

procedure

1.

Set the compass to a set length. Draw an arc around the point so that it intersects the line at two distinct points.

2.

At each of those points, draw an arc. This will result in two more arcs: one from each point. Make sure the length is set such that the two arcs intersect.

3.

Draw a straight line from the given point to the point where both arcs meet. This is the perpendicular line.

Or, presented visually:

Perpendicular to a line from a point on the line

This construction is very similar to the above construction, but this time, the point is on the line.

procedure

1.

Set the compass to a set length. Draw an arc around the given point so that it intersects the line in two points.

2.

At each of these points, set the compass to a slightly longer length and draw an arc. This will result in two more arcs: one from each point. Make sure the length is such that the two arcs intersect.

3.

Draw a straight line from the given point to the point where both arcs meet. This is the perpendicular line.