Construction – JSS3 Mathematics Lesson Note (Third Term)

Geometric construction is the accurate drawing of lines, angles and shapes using only a ruler and a pair of compasses.

Behavioural objectives

By the end of the lesson, students should be able to:

1. state the points to remember when using a ruler and compass;
2. construct the perpendicular bisector of a line segment;
3. bisect an angle;
4. construct angles of 60° and 30°;
5. construct an angle of 90°.

What are the key terms in construction?

Construction uses a few terms that you must know before you start.

• Line segment: a part of a straight line with two end points, such as XY.
• Arc: a part of a circle, drawn with the compass.
• Vertex: the point where two arms of an angle meet.
• Bisect: to cut into two equal parts.
• Midpoint: the point that divides a line segment into two equal parts.
• Perpendicular: meeting another line at an angle of 90°.
• Radius setting: the distance between the compass point and the pencil tip.

What instruments do you need for construction?

You need a hard pencil, a ruler and a pair of compasses.

What should you remember when using a ruler and compass?

Accurate construction depends on sharp tools, a steady compass and clear crossing points.

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. A blunt tip can slip during construction.
4. Set the compass and ruler as carefully as you can, so that your measurements are accurate.
5. Align your straight edge carefully along the line you intend to draw.
6. When constructing triangles, make sure the arcs cross at a clear point.
7. Do not rub out your arcs. They show how you made the construction.

How do you construct the perpendicular bisector of a line segment?

The perpendicular bisector of a line segment is a line that cuts the segment into two equal parts at right angles. Every point on it is the same distance from the two ends of the segment.

To construct the perpendicular bisector of the line segment XY:

1. Draw the line segment XY.
2. Open your compass to more than half the length of XY. Place the spike at X and strike two half-arcs, one slightly above and one slightly below the line.
   
3. Without adjusting your compass, put it on Y and draw another arc above and below the line.
   
4. Label the points where the arcs cross A and B.
   
5. Draw a straight line through A and B.
   
6. The point M where the line AB crosses XY is the midpoint of XY, and AM is perpendicular to XY.
   

How do you bisect an angle?

To bisect an angle is to divide it into two equal angles with a straight line called the angle bisector.

Let V be the vertex of the angle you want to bisect.

1. Place your compass on V and draw an arc that crosses both sides of the angle.
   
2. Label the crossing points A and B.
   
3. Place your compass on A and draw an arc between the two sides of the angle.
   
4. Without adjusting your compass, place it on B and strike another arc that cuts the one you have drawn from A. Label the point where the arcs cross C.
   
5. Draw a straight line through V and C.
6. The line VC bisects the angle. Angles AVC and BVC are equal.
   

How do you construct a 60° angle?

A 60° angle is constructed by making an equilateral triangle, because each angle of an equilateral triangle is 60°.

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 it at B and draw an arc that intersects the arc drawn through A at P.
4. Join A to P. The angle PAB is 60°.

Why the method works:

• Equal sides: AB, AP and BP are all the same length, since the compass setting did not change.
• Equilateral triangle: triangle ABP has three equal sides, so each angle is 60°.

How do you construct a 30° angle?

A 30° angle is constructed by first constructing 60° and then bisecting it, since half of 60° is 30°.

1. Draw the arm AB.
2. Place the point of the compass at A and draw an arc that passes through B.
3. Place the point of the compass at B and draw an arc that cuts the arc drawn in Step 2 at C. Angle CAB is 60°.
4. Keep the same compass setting. With the point at B, draw an arc near C, inside the angle.
5. With the point of the compass at C, draw an arc to cut the arc drawn in Step 4 to give point F.
6. Join F to A. The angle FAB is 30°.

How do you construct a 90° angle?

A 90° angle is constructed by making 120° with two 60° arcs and then bisecting it, or by bisecting a straight angle (180°).

To construct a 90° angle at P on the line PQ:

1. Draw the line PQ.
2. Place the point of the compass at P and draw a long arc that passes through Q and rises above the line.
3. Keep the same compass setting. With the point at Q, draw an arc that cuts the first arc at R. Angle QPR is 60°.
4. Keep the same setting. With the point at R, draw an arc that cuts the first arc at S. Angle QPS is 120°.
5. Bisect angle RPS: place the compass at R, then at S, with the same setting, and draw two arcs that cross at T.
6. Join P to T. The angle QPT is 90°.

Why the method works:

• Angle QPR: 60°.
• Angle RPT: half of 60°, which is 30°.
• Angle QPT: 60° + 30° = 90°.

Another method:

• Straight angle: extend the line through P, then bisect the straight angle at P as you bisect any angle. Each half is 90°.

How are the angle constructions related?

The 30° and 90° constructions both start from the 60° construction.

What mistakes should you avoid in construction?

Most errors come from changing the compass setting or drawing faint, unclear arcs.

• Changed setting: do not alter the compass opening between two arcs that must be equal.
• Arcs too short: arcs that do not cross cannot give a point. Draw them long enough.
• Thick lines: a blunt pencil gives thick lines and a wrong crossing point.
• Slipping point: hold the compass by its top and keep the spike firm.

Evaluation

1. Why should you use a hard pencil with a fine point for construction?
2. Describe how to construct the perpendicular bisector of a line segment XY.
3. List the steps for bisecting an angle with vertex V.
4. Explain why a 30° angle can be constructed by first constructing a 60° angle.
5. Explain why triangle ABP is equilateral in the construction of a 60° angle.

Assignment

Using a ruler and compass only, do the following in your notebook:

1. Draw a line segment 8 cm long and construct its perpendicular bisector.
2. Construct an angle of 60° and bisect it to obtain an angle of 30°.