**Geometrical Construction**

Using ruler and compasses: Remember the following when making geometrical constructions.

1. Use a hard pencil with a sharp point. This gives thin lines which are more accurate.

2. Check that your ruler has good straight edge. A damaged ruler is useless for construction work.

3. Check that your compasses are not too loose. Tighten loose compasses with a small screw driver.

4. All construction lines must be seen. Do not rub out anything which leads to the final result.

5. Always take great care, especially when drawing a line through a point.

6. Where possible, arrange that the angles of intersection between lines and arcs are about 90^{0}.

**Perpendicular bisector of a line segment**

The locus of a point which moves so that it is an equal distance from two points, A and B, is the perpendicular bisector of the line joining A and B.

**Perpendicular** means **at right angles to**.

**Bisector** means **cuts in half**.

To construct this locus, you do the following (try this yourself on a piece of paper):

Draw the line segment XY.

Put your compass on X and set it to be over half way along the line. Draw an arc.

Without adjusting your compass put it on Y and draw another arc.

Label these points A and B.

Draw a straight line through A and B.

The point M where the lines cross is the midpoint of XY. And AB is perpendicular to XY.

Bisecting an angle

V is the vertex of the angle we want to bisect.

Place your compass on V and draw an arc that crosses both sides of the angle.

Label the crossing points A and B.

Place your compass on A and draw an arc between the two sides of the angle.

Without adjusting your compass place it on B and draw another arc that cuts the one you just drew. Label the point where they cross C.

Draw a straight line through V and C.

The line VC bisects the angle. Angles AVC and BVC are equal.

**Constructing a 90 ^{0} Angle**

We can construct a 90º angle either by bisecting a straight angle or using the following steps.

**Step 1:** Draw the arm *PA*.

**Step 2:** Place the point of the compass at *P* and draw an arc that cuts the arm at *Q*.

**Step 3:** Place the point of the compass at *Q* and draw an arc of radius *PQ* that cuts the arc drawn in Step 2 at *R*.

**Step 4:** With the point of the compass at *R*, draw an arc of radius *PQ* to cut the arc drawn in Step 2 at *S*.

**Step 5:** With the point of the compass still at *R*, draw another arc of radius *PQ* near *T* as shown.

**Step 6: **With the point of the compass at *S*, draw an arc of radius *PQ* to cut the arc drawn in step 5 at *T*.

**Step 7:** Join *T* to *P*. The angle *APT* is 90º.

**Constructing a 30 ^{0} Angle**

We know that: ½ of 60^{0 }= 30^{0}

So, to construct an angle of 30º, first construct a 60º angle and then bisect it. Often, we apply the following steps.

**Step 1:** Draw the arm *PQ*.

**Step 2:** Place the point of the compass at *P* and draw an arc that passes through *Q*.

**Step 3:** Place the point of the compass at *Q* and draw an arc that cuts the arc drawn in Step 2 at *R*.

**Step 4:** With the point of the compass still at *Q*, draw an arc near *T* as shown.

**Step 5:** With the point of the compass at *R*, draw an arc to cut the arc drawn in Step 4 at *T*.

**Step 6:** Join *T* to *P*. The angle *QPT* is 30º.

**Constructing a 60 ^{0} Angle**

We know that the angles in an equilateral triangle are all 60º in size. This suggests that to construct a 60º angle we need to construct an equilateral triangle as described below.

**Step 1:** Draw the arm *PQ*.

**Step 2:** Place the point of the compass at *P* and draw an arc that passes through *Q*.

**Step 3:** Place the point of the compass at *Q* and draw an arc that passes through *P*. Let this arc cut the arc drawn in Step 2 at *R*.

Step 4: Join P to R. The angle QPR is 60^{0}, as the ∆PQR is an equilateral triangle.

Try your understanding regarding the explanations above over and over again.