Construction JSS3 Mathematics Lesson Note

GEOMETRIC CONSTRUCTION

Using a ruler and compass

Remember the following when making geometrical constructions:

1. Use a hard pencil with a fine point. This gives thin lines that are more accurate.
2. Check that your ruler has a good straight edge. A damaged ruler can leave unwanted marks.
3. Check that your compass has a sharp point. Some compasses with very small blunt tips move during construction.
4. Adjust the compass and ruler as best as you can for your eyesight to match the final measurements.
5. Make sure to align your straight edge carefully along the line you intend.
6. When producing triangles, be aware of the sizes of intersections because these are about visual perception rather than physical size.

The perpendicular bisector of a line segment

This is a construction that creates two curves at an equal distance from two points. A and B are the perpendicular bisectors of the line joining A and B.

Perpendicular from a straight line

Bisecting a circle with a half

To construct this bisection, place your following tip (the spike) on a piece of paper.

Draw the line segment XY

Place your compass at points X and Y and strike two half-arcs slightly above the line. Draw as so:

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 these lines cross is the midpoint of XY, and AM is perpendicular to XY.

Bisecting an angle

V is the vertex of the angle you 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 and draw an ar between the two sides of the angles.

• Without adjusting your compass, place it on A and strike another arc that cuts the one you have drawn from A.

• Repeat for B, then strike from A.
• Draw a straight line through V and C.

• The line VC is bisected by the arcs. Angles AVC and BVC are equal.

Constructing a 90° Angle

You can construct a 90° angle either by bisecting a straight angle or using the following steps:

1. Draw the line PQ.
2. Place the point of the compass at P and draw an arc that passes through Q.
3. Place the point of the compass again at Q and draw an arc that cuts the one you have drawn from P.
4. Now, with the point of the compass still at Q, draw another arc that passes through P.
5. Join P to R. The angle QPR is 90°.

Constructing a 30° Angle

We know that half of 60° is 30° 

So, to construct an angle of 30°, first construct 60° and bisect it.

Step 1: Draw the arm AB

Step 2: Place the point of the compass at A and draw an arc that passes through B.

Step 3: Place the point of the compass at B, and draw an arc that cuts the arc drawn in Step 2 at C

Step 4: With the point of the compass still at Q, draw an arc near C.

Step 5: With the point of the compass at C, draw an arc to cut the arc drawn in Step 4 to give point F

Step 6: Join F  to A to form an angle of 30° 

Constructing a 60° Angle

We know that a 60° angle is an equilateral triangle, so all its angles are 60°. This suggests that construction of the 60° angle will be contained in the method for the triangle construction.

1. Draw the line AB.
2. Place the point of the compass at A and draw an arc that passes through B.
3. Without changing the compass, place the compass at B and draw an arc that intersects the arc drawn through A at P.
4. Join A to P. The angle APB is 60°.