# Construction and Loci Problems

## Topic Overview

In the field of mathematics, ‘construction’ means to draw shapes, angles or lines accurately.

These constructions require you to use a compass, a ruler and a pencil. For example, during your GCSE maths exam, you may be required to construct a triangle. There are two methods of drawing triangles; with a compass or by using a ruler and protractor.

*Example*

(a) - Construct a triangle with side lengths of 3cm, 4cm and 5cm.

*Solution*

(a) - Method One - Using a compass

Firstly, use your ruler to draw a 5cm line. Label either end of this line point A and point B.

Secondly, open your compass to a radius of 4cm and place the compass needle at point A. Using this radius of 4cm draw a arc.

Then, place your compass on point B and draw a similar arc. Find the point where these two arcs cross and join each end of the line to this point where they cross.

*(Note: Do not erase any construction arcs when using this method, in order to demonstrate your method to the examiner who is marking your paper).*

*Example*

(b) - Construct a triangle which has one side length of 6cm, and two angles of 55° and 70°.

*Solution*

(b) - Method Two- Using a ruler and protractor
When you are asked to construct a triangle and the question tells you the size of some of its angles, it is often easier to use a ruler and protractor as opposed to a compass.

Firstly, use a ruler to draw a 6cm and label either end of this line point A and point B.

Secondly, place the centre of your protractor on point A and line up your protractor at 0°. From this point, measure an angle of 55° and mark it with a dot. Connect this dot to point A with a ruler.

Similarly, place your protractor on point B, line it up with 0°, measure an angle of 70° and mark it with a dot. Connect this dot to point B with a ruler and it should meet with the line you have drawn from point A.

A 'locus' is a path which is formed by a point which moves according to a specific principle or rule. The plural of locus is 'loci', and is used to describe a set of all points which share a property. When connected, these points will create a circle. For example, if you connect all of the points on the grid below, you will create a circle:

## Key Concepts

In the new linear GCSE Maths paper, you will be required to solve various construction problems. The specific questions you will be expected to answer will vary depending upon which examination board with which you are registered, but as a rule you will be required to:

- Use a straight edge and a pair of compasses to draw constructions
- Construct loci and solve mathematical problems using loci

Listed below are a series of summaries and worked examples to help you solidify your knowledge about constructions and loci.

## Worked Examples

**1 - Perpendicular bisector of a line segment**

During your GCSE maths exam, you may be required to construct the perpendicular bisector of a line segment.

The **'perpendicular bisector of a line segment'** is the locus of a point which moves in an equal distance from two points; A and B.

**'Perpendicular'** is the mathematical term used to describe lines which are at right angles to one another.

**'Bisector'** is a mathematical term meaning 'to cut in half'.

*Example*

(a) - Construct the perpendicular bisector to the line AB.

*Solution*

(a) - In order to construct this locus, you must follow these steps:

- Firstly, draw a straight horizontal line and label either end as points A and B respectively.
- Secondly, place your compass on point A, and extend your compass so that it is positioned halfway along the line. Using your compass, draw an arc along this halfway point.
- Whilst keeping your compass extended to this length, position your compass on point B and draw an arc halfway along the line. Locate the two points where these arcs intersect and label them as points C and D.

- Draw a straight line through these points C and D and you will have constructed the perpendicular bisector to the line AB.

*(Note: You can check that your answer is correct by measuring the angle between line AB and line CD. If this angle is 90 °, then the two lines are perpendicular).*

**2 - How to bisect an angle**

If you are asked to 'bisect an angle', this means that you are being asked to 'cut the angle in half'. You can bisect an angle by following this method:

*Example*

(a) - Bisect the angle PQR

Solution

(a) - Firstly, place your compass on the point Q and draw an arc which crosses both sides of the angle. Label these points A and B.

Secondly, place your compass on point A and draw an arc between the two sides of the angle. Without changing the extension of your compass, place it on point B and draw an arc between the two sides of the angle. Locate the point where these two arcs cross and label it point C.

Finally, draw a line through point Q and point C. This line QC will bisect the angle in question.

*(Note: In order to check your answer is correct, measure the two angles either side of line XC. If you have correctly bisected the angle, these two angles will be of equal size).*

**3 - Solving loci problems **
As mentioned in the topic overview, when a point moves in a plane according to a series of given conditions, the path along which it moves is referred to as a 'locus'. The term 'loci' is the plural of 'locus'.

During your GCSE maths exam, you may be required to solve various problems involving loci:

*Example*:

(a) - Construct the locus of a point P at a constant distance of 3cm from a fixed point Q.

*Solution:*

(a) - In order to construct this locus, measure and draw a 3cm line. Label either side of the line as points P and Q respectively.

Extend your compass to 3cm and place the needle of your compass on point Q. Draw a full circle around point Q. This circle should cross point P:

*Example*

(b) - Two cows graze within a field which has a length of 20m and a width of 12m. The cows are penned within diagonally opposite corners of the field, which have lengths of 13m.
On a drawing which is sketched to scale, demonstrate the sections of the field which are grazed by both cows.

**Solution**

(b) - The question has stated that your drawing must be sketched to scale. An appropriate scale would be 2m = 1cm. As a result, you must adapt your measurements to fit this scale:

20m = 10cm , 12m = 6cm , 13m = 6.5cm

Once you have set the scale, you need to measure a distance of 6.5cm using your compass. To do this, place your compass point on one corner of the rectangle, and draw an arc. Repeat this process for the diagonally opposite corner.

As a result, the area in between these 2 arcs will be grazed by both cows. It is helpful if you shade in this area where both cows graze:

## Exam Tips

- Always sharpen your pencil before constructing shapes and loci. This will make your answers more accurate.
- Double check all of your constructions by measuring the angles and lengths you have constructed.
- Do not erase any of your construction markings because they will earn you valuable method marks during your exam.
- Remember that all the loci of a particular point will form a circle once connected. This rule can be used to check that you have correctly constructed particular loci.

## Topic Summary

The GCSE maths topics of 'Construction Problems and Loci' is highly dependent upon accuracy. As a result, you need to make sure you bring the correct equipment into the examination with you. Remember to bring:

- A compass
- A ruler
- A protractor
- A sharpened pencil (it is also advisable that you bring a spare pencil)
- A pencil sharpener
- A rubber

If you bring this necessary equipment, you can correctly sketch accurate shapes, loci and bisect angles. Moreover, sketching constructions and solving loci problems is greatly aided by practice. Therefore, if you attempt as many past paper questions as possible, you will become confident at this topic and earn the maximum number of marks in your actual GCSE exam!

## Related Topics

- Polygons
- Congruent Triangles
- Trigonometry and Pythagoras
- Perimeter, Area, Volume
- Transformations
- Circle Theorem Problems
- Proof