## What you need to know

## Loci and construction – The four rules

In GCSE maths, there are **4 ruler and compass** constructions you must to know.

**Locus of all points**around a point or line**Perpendicular bisector****Perpendicular line**from a point.**Bisecting an angle**.

**Locus of all points around a point or line **

A **locus**(**loci **is the plural) is a collection of points which share a property.

E.g. The circumference of a circle is the locus of all points in 2D that are the same distance from a particular point – the centre.

**Example:** Draw the locus of all the points 1cm from line AB

– First we set the compasses to 1cm, and draw a half circle around point A

– Next keeping the compass the **same size**, repeat for point B

– Finally using a ruler join up the two circles.

All these steps are shown in the diagram in **red**.

**Perpendicular bisector**

The first construction is a **perpendicular bisector **of a line.

**Example:** Bisect line AB show.

(or it could be asked as, *draw a line equidistant from points A and B*, the method will be exactly the same_

To do this you must do the following:

– Set your compasses to a fixed length apart (**Must be greater then half the line**)

– Put your compass on point A and draw an arc (blue).

– With your compasses at the **same length**, repeat step 2 for the other end of the line (also blue).

– Then, draw a line which passes through the two crossing points. This line (**red**) is the perpendicular bisector.

## Perpendicular Line From a Point.

The second construction is a **perpendicular from a line to a point**.

In the diagram, we have been given a line segment and a point (both black).

**Example:** Construct a perpendicular line from point C to line AB

– Place your compass on the point C and draw an arc of a circle that passes through the line twice (**blue**).

– Place your compass on the crossing point (**green cross**) and draw a small arc on the opposite side of the line to where the point is (green).

– With your compasses at the **same length**, repeat step 2 for the other crossing point (other green cross).

– Draw a line that passes through the original point and the point where the last two arcs cross.

This line (**red**) is the perpendicular from the line to the point.

## Bisecting an angle

The third construction is an **angle bisector**. This allows us to split a given angle perfectly in half.

**Example:** Construct a line, equidistant from line AB and BC

In the diagram, we have been given two lines (black) and an angle between them. To bisect it, we do the following:

– Place your compass on the corner where the two lines meet and draw an arc (**blue**) that passes through both lines.

– Place your compass on the crossing point (**green cross**) and draw a small arc (**green**) between the lines.

– With your compasses at the **same length**, repeat step 2 from the other crossing point.

– Draw a line (red) passing through the corner where the lines meet and the point where the two green arcs cross. This is the angle bisector.

**Example**

Points A and B are 6cm apart.

Shade the locus of points that are closer to point A than point B but less than 4cm from point B.

Firstly, we need to find the points that are **“closer to point A than point B”**. This means all the points that are to the left of the line which falls halfway between A and B. This means we need to draw a **perpendicular bisector**.

Following the steps, detailed above, we get the perpendicular bisector of line AB shown by the orange line.

Next all points are less than 4cm from B, for this we use the **locus of points around a point**.

We set out compasses to a length of 4cm, placing the needle on B we draw our circle.

Finlay we shade all the points beyond the orange line but closer then 4cm from B.

This can be seen in **green**on the diagram.

### Example Questions

**Question 1: **Draw the locus of points that are equidistant from lines AB and BC. (Hint: which one of the 4 constructions does this look like?)

You may have noticed that this is reminiscent of that time we sliced an angle in half, and as it happens, we’re going to do that again. The line that bisects the angle is also halfway between the two lines that form the angle – it must be, otherwise it wouldn’t cut the angle in half.

So, we place the compass at B and draw an arc that passes through both lines.

Then, placing the compass at each of the points where the first arc crossed, we draw two small arcs which lie in between lines AB and BC.

The line which passes through the corner at B and the point where the latter two arcs cross is the angle bisector, aka the locus of points which are equidistant from AB and BC. The correct picture should look like the one below (construction lines in blue, locus in red):

**Question 2:** Draw the locus of all points which are 1cm away from the line below. (HINT: imagine a 1cm string that is attached at one end to the line and can move along it – what are the furthest reaches of that string?)

This one doesn’t require any of the 4 constructions from this topic, but you still need a compass and a ruler.

We will break this into parts. Firstly, the locus of points which are 1cm directly above the line is another line, 1cm away from, and parallel to, this line. Same goes for the locus directly below the line.

When we reach the ends of the line, it’s a little different. The locus changes from being a straight line to a curve. To understand this, imagine standing at one end of a line and reaching your arm out 1m in all directions. The shape that your reach forms is a curve, specifically a semi-circle – this is the locus, since your reach extends the same distance, 1m, in every direction.

In short, the ends of the circle act like points, with their loci being circles (or semi-circles). The resulting picture should look like the one below (with the locus in red).

**Question 3:** Bill is planning to put a fountain in his garden. His house is situated along the line CD, and there is a tree at point E. He wants the fountain to be at least 3m away from his house and at least 1.5m away from the tree. Shade the locus of points where he could have the fountain.

1m = 1cm.

So, for the fountain to be at least 3m away from his house along CD, we need to only consider the area to the left of the straight line which is parallel to CD and 3cm away from it.

Then, the locus of points which are 1.5m away from the tree at E will be a circle of radius 1.5cm – for the fountain to be at least 1.5m away, it must be outside this circle.

So, the locus of points where he could place the fountain is to the left of the (blue) line 3m away from the house, and outside the (green) circle which is 1.5m away from the tree. The correct region is shaded red on the picture below.

**Question 4: **Define the bisector of an angle.

The bisector of an angle is a line segment which divides the angle into two equal parts.

**Question 5: **Construct a line perpendicular to the line AB that passes through the point P, shown below.

Construction of a line perpendicular to AB passing through point P as shown:

### Worksheets and Exam Questions

#### (NEW) Loci & Constructions Exam Style Questions - MME

Level 4-5#### Loci And Construction 1 - Drill Questions

Level 4-5#### Loci And Construction 2 - Drill Questions

Level 4-5### Videos

#### Loci and Construction Q1

GCSE MATHS#### Loci and Construction Q2

GCSE MATHS#### Loci and Construction Q3

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