GCSE Maths Geometry and Measure

Loci and Construction

Constructions

# Constructions

Here we will learn about constructions, including how to draw accurately using a pencil, a ruler and a pair of compasses.

There are also constructions worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What are constructions?

Constructions are accurate drawings of shapes, angles and lines in geometry.

To do this we need to use a pencil, a ruler (a straight-edge) and compasses.

The basic constructions are perpendicular bisector and angle bisector.

Perpendicular bisector Angle bisector Step by step guide: Perpendicular bisector (coming soon)

Step by step guide: Angle bisector (coming soon)

Perpendiculars can also be constructed for a point and a given line.

Perpendicular from a point to the line Perpendicular with a point on
the line Step by step guide: Constructions between points and lines (coming soon)

Other geometric constructions include how to draw a regular hexagon.

Step by step guide: How to draw a hexagon (coming soon)

The compass constructions can be applied on the same diagram. For example a 60 degree angle can be constructed and used to construct a 30 degree angle.

Similarly a 90 degree angle can be constructed and used to construct a 45 degree angle.

Step by step guide: How to construct a 30, 60, 45, 90 degree angle (coming soon)

### What are constructions? ## How to construct a perpendicular bisector

In order to construct the perpendicular bisector of a line segment:

1. Use compasses to draw an arc.
2. Use compasses to draw a second arc, intersecting the first arc.
3. Join the 2 points where the arcs intersect.

### Explain how to construct a perpendicular bisector ## Constructions examples

### Example 1: perpendicular bisector

Construct a perpendicular bisector of the given line AB:

1. Use compasses to draw an arc.

Open the compasses to about three-quarters of the length of the line.

Put the point of the compasses on one of the endpoints of the line.

Draw an arc.

2Use compasses to draw a second arc, intersecting the first arc.

Keeping the compasses the same, draw another arc from the other end of the line.

3Join the two points where the arcs intersect.

Using a straight-edge (a ruler), join up the two points where the arcs intersect each other.

The new line is the perpendicular bisector of the original line segment AB.

You can check that the new line goes through the midpoint of the line segment AB by using a ruler to measure. The line AB should have been cut into two equal halves at a right angle.

### Example 2: angle bisector

Construct an angle bisector of angle ABC:

Set your compasses to a length that is less than the shortest arm.  Putting the point of the compasses on B, draw one arc going through both AB and BC.

Put the point of the compasses on the point where the first arc crossed AB and draw an arc. Keep the compass on the same setting. Repeat by putting the point of the compasses on the point where the first arc crossed BC and draw an arc. These two arcs need to intersect.

Using a straight-edge – a ruler, join up the point where the arcs intersect each other with the vertex B.

The new straight line is the angle bisector of the original angle ABC.

You can check that the new straight line bisects the angle ABC by using a protractor to measure.  The angle ABC should have been cut into two equal angles. The two new angles are congruent.

### Example 3: from a point to a line

Construct a line from point P perpendicular to the line.

Put the point of the compasses on the original point P.

Draw an arc that crosses the original line in two places.

Put the point of the compasses on a point where the arc crosses the line and draw an arc.  Keep the compasses on the same setting.  Repeat with the second point where the arc crosses the line; drawing another arc to intersect the arc just drawn.

Using a straight-edge (a ruler), join up the point where the arcs intersect each other and the original point P.

The new line is perpendicular to the original line segment. You can check if the lines meet at a right angle.

### Example 4: from a point on a line

Construct a perpendicular line passing through given point P.

Put the point of the compasses on the original point P.

Draw an arc that crosses the original line in two places.

The compasses will need a wider setting. Put the point of the compasses on a point where an arc crosses the line and draw arcs on both sides of the line. Repeat with the other point where an arc crosses the line; drawing another pair of arcs to intersect the arcs just drawn.

Using a straight-edge (a ruler), join up the two points where the arcs intersect each other. The new line should go through the original point.

The new line is perpendicular to the original line segment. You can check if the lines meet at a right angle.

### Example 5: draw a hexagon

Construct a hexagon:

Use compasses to draw a circle and mark the centre point with a cross.

Keep the compasses set the same and make a starting mark and then draw arcs. Keep moving the compasses on until there are 6 marks on the circumference of the circle.

With a straight-edge draw lines between the points where the arcs cross the circle.

### Example 6: construct a 60 degree angle

Construct a 60^o angle:

With a pencil and a straight-edge draw a straight line.

Use compasses centered on one end of the line, draw an arc.

Keep the compasses at the same setting. The new arc should cross the first arc.

Use a straight-edge to join the end point used in step 2 to the intersection of the two arcs.

The angle should be 60^o. You can use a protractor to check.

### Example 7: construct a 30 degree angle

Construct a 30^o angle:

Follow the steps in Example 6 to construct a 60 degree angle.

Angle ABC is 60^o .

Place the point of your compasses at A and draw an arc. Keep the compasses at the same setting and repeat placing the point at C and drawing another arc. Using a straight-edge, join up the point where the arcs intersect each other with the vertex B.

The 60 degree angle has been bisected into two equal angles, both 30 degrees. You can check by using a protractor to measure.

### Example 8: construct a 90 degree angle

Construct a 90^o angle:

.Set your compasses to about three-quarters of the length of your line. Place the point of your compasses on one endpoint of the line and draw an arc.

Keeping the compasses set at the same setting, draw another arc from the other endpoint. Finally use a straight-edge to join the two intersections of the arcs.

The new line is the perpendicular bisector of the original line segment.

### Example 9: construct a 45 degree angle

Construct a 45^o angle:

Follow the steps in Example 8 to construct a 90 degree angle.

Place the point of your compasses at the centre and draw an arc. Place the point of your compasses at the intersection and draw an arc. Keep the compasses at the same setting and repeat.

Place the compasses at the other intersection and draw another arc. Using a straight-edge, join up the point where the arcs intersect each other with the centre.

The 90 degree angle has been bisected into two equal angles, both 45 degrees. You can check by using a protractor to measure.

### Example 10: all three sides

Draw the triangle ABC accurately:

Use a pencil and a ruler to draw the base. It is usually easiest to use the longest side.

Open the compasses to 5 cm . Place the point on point A and draw an arc.

Open the compasses to 3 cm . Place the point on point B and draw an arc. This second arc should cross the first arc.

Complete the triangle by joining the point where the arcs intersect to point A and point B.

### Example 11: two angles and the included side

Draw the triangle ABC accurately:

Use a pencil and a ruler to draw the base.

At point A use a protractor to measure the angle 50^o , make a mark and then draw a straight line from point A through the mark. Make this line long.

At point B use a protractor to measure the angle 35^o , make a mark and then draw a straight line from point B though the mark.

Make sure that the two lines interact to form the triangle.

### Example 12: two angles and the included side

Draw the triangle ABC accurately:

Use a pencil and a ruler to draw the base.

At point B use a protractor to measure the angle 40^o , make a mark and then draw a straight line from point B through the mark. Make this line long.

Open the compasses to 5 cm . Place the point on point B and draw an arc. This second arc should cross the line drawn in step 2 .

Join the intersection of the arc and the line with point A.

### Common misconceptions

• The construction arcs must not be removed

The arcs drawn should be drawn lightly so can be adjusted if needed but they must be visible in your final answer. This is to show that you have used the correct method to draw the perpendicular bisector accurately.

• The pencil should be sharp

A sharp pencil helps your diagram to be accurate. Using a small pencil in compasses can also be helpful.

### Practice constructions questions

1. Construct a perpendicular bisector of the line MN.         The construction must be made with compasses kept to the same setting for both arcs. The construction arcs must be seen. The perpendicular should be drawn with a straight edge.

2. Construct an angle bisector of XYZ         The construction must be made with compasses kept to the same setting for the arcs.  The construction arcs must be seen. The line should be bisecting the original angle Y.

3. Construct a perpendicular from point P on the line:         The construction arcs should be made with compasses and be visible. The final line should go through the original point and be drawn with a straight-edge.

4. Construct an isosceles triangle with sides 8 cm, 6 cm and 6 cm        The longest side should be 8 cm . Compasses should be used to make the arcs, and they should be clearly seen. The other two sides are the same length; 6 cm.

5. Construct a triangle ABC, with AB is 6 cm , angle BAC is 60^o and angle ABC is 45^o.        Check the labelling of the triangle. The side AB should be 6 cm. The angle at A should be 60^o and angle at B should be 45^o.

6. Construct a triangle PQR, with PQ is 6 cm , QR is 5 cm and angle PQR is 60^o.        Check the labelling of the triangle. The side AB should be 6 cm . The angle at A should be 60^o and angle at B should be 45^o.

### Constructions GCSE questions

1. Use a ruler and compasses to construct the perpendicular from point P to the line AB. You must show your construction lines. (2 marks) for the first arc(s) crossing the line centered on P.

(1)

for the perpendicular with all construction arcs.

(1)

2. In the space below, use a ruler and compasses to construct an equilateral triangle with sides 6 cm.

You must show all your construction lines.

One side of the triangle has already been drawn for you. (2 marks) for the construction arcs.

(1)

for the completed equilateral triangle.

(1)

3. The diagram shows the plan of a garden. A statue in the garden is equidistant from PS and from SR.

The statue is exactly 6 m from Q.

On the diagram, mark the position of the statue with a cross (\times).

(3 marks) for the angle bisector of PSR.

(1)

for arc of circle radius 6 cm centre Q.

(1)

for the correct point.

(1)

## Learning checklist

You have now learned how to:

• Construct a perpendicular bisector
• Construct an angle bisector
• Construct a perpendicular to a line including a point
• Construct a hexagon
• Construct triangles
• Construct a 30, 60, 45, 90 degree angle

## Still stuck?

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