Year 8 Interactive Maths - Second Edition

## Bisecting Intervals and Angles

Bisect means to cut in half.  In this section, we will consider the methods to bisect an angle and a line segment.

### Bisector of an Angle

The steps required to bisect (cut in half) an angle are shown in the following example.

#### Example 3

Use a ruler and compass to bisect the angle ABC:

##### Solution:

Step 1:  Draw an arc with B as the centre to cut the arms, BA and BC, of the angle at P and Q respectively.
Step 2:  Using the same radius, draw an arc centred at P.
Step 3:  With centre Q and using the same radius, draw an arc to cut the arc in Step 2 at R.
Step 4:  Join, B, the vertex of the angle to the point R.

BR bisects the angle ABC, and is called the bisector of angle ABC.

### Perpendicular Bisector of a Line Segment

The steps required to construct a perpendicular bisector of a line segment are shown in the following example.

#### Example 4

Use a ruler and compass to perpendicularly bisect a line, AB, 8 cm long.

##### Solution:

Step 1:  Draw a line, AB, 8 cm long.
Step 2:  Draw an arc centred at A with a radius of more than half of the length of AB.
Step 3:  Using the same radius, draw an arc centred at B to cut the arc drawn in Step 2 at P and Q.
Step 4:  Join PQ.

PQ perpendicularly bisects AB at M; and PQ is called the perpendicular bisector of the line segment AB.

### Perpendicular from a Point onto a Line

#### Example 5

Draw a line, AB, 7 cm long.  Then use a compass to drop a perpendicular from a point, P, which is approximately 6 cm from the line.

##### Solution:

Step 1:  Draw a line, AB, 7 cm long.
Step 2:  Draw an arc centred at P with a radius that is long enough to cut the line AB at two points, X and Y.
Step 3:  Draw an arc centred at X using a radius that is more than half of the length of the interval XY. You could use the radius PX.
Step 4:  Using the same radius, draw an arc centred at Y to cut the arc drawn in Step 3 at Q.
Step 5:  Join PQ.

### Perpendicular at a Point on a Line

#### Example 6

Draw a line, AB.  Then use a compass to construct a perpendicular to the line AB at a point, P, on the line.

##### Solution:

Step 1:  Draw a line, AB, and mark the point P on the line.
Step 2:  Draw an arc centred at P that cuts the line AB at two points, X and Y.
Step 3:  Draw an arc centred at X; using a radius that is longer than half of the length of XY.
Step 4:  Using the same radius, draw an arc centred at Y to cut the arc drawn in Step 3 at Q.
Step 5:  Join PQ.