Locus and Geometric construction

Constructing lines

How to construct a line of given dimensions

To construct a line AB of given dimensions, say 8cm long, you first draw a line of a random length that is more than 8cm. Then mark your starting point as A, as shown in the diagram below:

Construct a line

After drawing the line, then you open your compass to a radius equal to the given length of the line, which is 8cm in this case. With the compass anchored at point A, scribe an arc to cut the line at a point you will mark as point B.

Construct a line

That is how you construct a line. My complete diagram is shown below. Do not erase the construction lines and arcs. It will result in loss of marks.

Construct a line


How to construct a perpendicular bisector on line AB

A perpendicular bisector of line AB is the locus of points equidistant from A and B. Here are the different ways you might be asked to construct a perpendicular bisector of a line:

  • bisect line AB
  • construct a perpendicular bisector of AB
  • construct an angle of 90° through the center of line AB

In any of those cases, the method is still the same. First we are going to assume you already have the line AB drawn. With the compass anchored at A, opened to a radius that is slightly more than half the line, scribe an arc above the line. Scribe another arc below the line using the same radius and the same anchor.

Bisect a line

Maintaining the same compass radius, anchor the compass at B and scribe two arcs, one above the line and the other below the line. The second set of arcs should cut the first set of arcs as shown below:

Bisect a line

Now draw a line through the intersection of the arcs. That is the perpendicular bisector of line AB.

Bisect a line

Constructing angles

In geometric construction, we are usually required to construct angles using a ruler and compasses only. It is possible to construct angles of 90°, 60° and their subsets as as we shall see.

How to construct an angle of 90°

An angle of 90° can be constructed in three different ways:

  • through the center of the line,
  • through a point on the line,
  • or from an external point to the line.

Constructing an angle of 90° through the centre of the line is the same as constructing a perpendicular bisector of a line. We already dealt with it, we are not going to look at it again. We are going to look at the other two methods.

How to construct an angle of 90° through point A on line AB

Open your compass to a reasonable radius (3 to 4cm approx) and, using point A as the anchor, scribe two arcs to cross the line, one behind the point and the other after the point. As shown below:

Construct 90

Those two points where the arc crosses the line will be our working points in constructing the 90° angle. Using the first crossing point as the anchor, scribe two arcs, one above the line and the other below the line. The radius of the compass should be slightly more than the distance between our two working points, as shown below:

Construct 90

Repeat the same procedure using the same radius and the second crossing point as the anchor. The second set of arcs should cross the first set of arcs as shown below:

Construct 90

Finally draw an arc through the intersections of the arcs as shown below:

Construct 90


How to construct an angle of 90° from an external point M to line AB

In this case you are given line AB and an external point M. The objective is to construct a line from M to cross the line AB at a right angle. We are going to start with the diagram shown below:

Construct 90

Using point M as the anchor, scribe an arc to cut the line at two points as shown below:

Construct 90

Using the same radius and the first crossing point as the anchor, scribe an arc below the line. Repeat the same procedure using the second crossing point as the anchor. The two arcs should intersect below the line as shown below:

Construct 90

Finally draw a line from M through the intersection of the two arcs.

Construct 90