In these lessons, we will learn

- how to construct a perpendicular to a line, through a point on a line.
- how to construct a perpendicular to a line, through a point not on a line.

We will now show how to construct a perpendicular to a line segment, through a point on the line.

*Example : *

Construct a line perpendicular to *PQ* through the point *R* on *PQ*.

* Solution: *

**Step 1: **Stretch your compasses a short length. It can be any length. Put the sharp end of the compasses on point *R* and mark two arcs *S* and *T* on the line *PQ.*

** Step 2:** Stretch the compasses a little more. Put the sharp end of the compasses on point *S* and make an arc with pencil end. Do the same on point *T* but make sure that the second arc intersects the first arc.

**Step3:** Draw a line from point *R* to the point where the arcs intersect. This line is perpendicular to *PQ* and passes through the point *R.*

1. Place compass point on the given point, and draw arc either side of point such that it intersects line twice.

2. Place compass on each arc-line intersection, and draw an arc above the line, without changing the compasses' width.

3. Using a straight edge, connect this intersection point to the original point with a straight line.

We will now construct a perpendicular to a line, through a point not on the line.

* Example* :

Construct a line perpendicular to the line *AB* and passing through the point *P, *that is not on the line,

* Solution: *

** Step 1:** Put the sharp end of your compasses on point *P,* stretch the compasses and mark two arcs *R* and *S* on the line *AB. *

Step2: Without changing the width of the compasses, put the sharp end of the compasses at *R* and draw an arc on the other side of *AB* compared with *P*. Do the same at *S* and make sure that the second arc intersects the first arc.

**Step 3 : **Draw a line from *P* to the point of intersection. This line passes through point *P* and is perpendicular to *AB*.

1. Place compass point on the given point, and draw arc such that it intersects line twice.

2. Place compass on each arc intersection, and draw a further pair of arcs which intersect each other on the other side of line from original point.

3. Using a straight edge, connect this intersection to the original point with a straight line.

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