**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*.
Note:
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*.
Activity 10.2
Key Terms
bisect, bisector of an angle, perpendicular
bisector of a line segment, perpendicular from a point onto a
line, perpendicular at a point on a line |