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Here we will learn about loci, constructions and bearings, including drawing various constructions, using constructions when drawing loci and measuring and calculating bearings.

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

**Loci and construction **are geometric objects that are drawn with specific features.

To draw loci and constructions **accurately**, you will be required to use a ruler, a pair of compasses, and a protractor when appropriate.

- A
**ruler**is used to**measure distances**and**construct straight lines**. - A
**pair of compasses**is used to draw points that are a**fixed distance from a point**. - A
**protractor**is used to**measure**and**construct angles**.

**Constructions** are set methods for creating accurate drawings in maths.

A **locus** of points (plural **loci**) is a set of points that follow a given rule. We often use constructions to accurately draw loci.

A** bearing** is an angle, measured clockwise from north, used to describe the position of an object. Bearings are given using **three figures**, for example 052^{\circ}.

There are a number of different constructions that you need to be able to perform for GCSE maths. These include,

- Angle bisector
- Perpendicular bisector of a straight line
- Perpendicular line to or from a particular point
- Triangles, given information about sides and/or angles
- Polygons

To do constructions we carefully follow the method, leaving all construction lines visible.

The most common constructions are,

When doing constructions, all construction lines should be left visible.

**Step-by-step guide:** Constructions

A **locus** of points (plural **loci**) is a set of points that follow a given rule. Given a rule, we need to be able to identify the points that follow that rule.

Here are some examples,

**Step-by-step guide:** Loci

A bearing is an angle, measured clockwise from north, used to describe the position of an object. They are written using three figures. Bearings can be drawn, measured or calculated depending on the problem.

The bearing of **B from A** is 075^{\circ}. This means to get from **A to B**, you need to travel in a direction 75^{\circ} clockwise from north.

We can find the bearing of A from B using angle rules.

As both north lines in the diagram below are in the same direction, these lines are parallel. This means that the angle at A and the angle at B are **co-interior** (their angle sum is 180^{\circ} ).

The angle x is equal to 180-75=105^{\circ}.

The **back bearing** is the bearing of **A from B**. As the sum of angles at a point is 360^{\circ}, the bearing \theta of A from B in the diagram below is 360-105=255^{\circ}.

**Step-by-step guide:** Bearings

Get your free loci and construction worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free loci and construction worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEBisect the angle ABC.

**With the point of the compasses on the vertex, draw an arc.**

2**With the point of the compasses on each intersection, draw two further arcs.**

3**Draw a line from the vertex to the intersection of the arcs.**

**Step-by-step guide:** Angle bisector

Draw the locus of points equidistant from lighthouse A and lighthouse B.

The locus of points equidistant from two points is the perpendicular bisector of the line between those points. Imagine a straight line connecting the two points A and B.

We need to construct the perpendicular bisector of the line AB.

**With the point of the compasses on one end of the line, draw an arc.**

Here we will place the point of the compass at A and draw an arc.

The radius of the arc must be greater than half of the distance between A and B.

**With the point of the compasses on the other end of the line, draw a second arc.**

Placing the point of the compass at B, we construct another arc with the same radius as the previous arc.

The radius of each arc must be the same, otherwise the two points of intersection will not be the same distance from A and B.

**Draw a straight line through the two points where the arcs cross.**

Using a ruler, draw a straight line through both pairs of intersecting arcs**.**

The line we have drawn is the locus of points equidistant from A and B.

**Step-by-step guide:** Perpendicular bisector

Two phone masts, R and S are 50 \ km apart. The signal from phone mast R can reach 30 \ km and the signal from phone mast S can reach 35 \ km. Using the scale 1 \ cm=10 \ km, shade the area covered by signal from both phone masts.

**Use the wording of the region required to decide what constructions are needed.**

In this case, we need to draw a circle with radius 3 \ cm (to represent 30 \ km ) around mast R and a circle with radius 3.5 \ cm (to represent 35 \ km ) around mast S.

**Perform any relevant constructions for points or line segments involved.**

For the first circle with centre R, measure the distance between the point of the compass and the pencil to 3 \ cm using a ruler. Place the point of the compasses on R and draw a circle. The radius of this circle is 3 \ cm.

For the second circle with centre S, measure the distance between the point of the compasses and the pencil to 3.5 \ cm using a ruler. Place the point of the compasses on S and draw a circle. The radius of this circle is 3.5 \ cm.

**Indicate the region required as necessary.**

The shaded area is covered by signals from both phone masts.

**Step-by-step guide:** Constructions between points and lines

Below is a diagram of Ellie’s garden. Ellie wants to plant a new tree. She wants the tree to be closer to the side AD than the side AB, more than 2 \ m from A and less than 3 \ m from the side AB. Using the scale 1 \ cm=1 \ m, indicate the area in which Ellie can plant the tree.

**Use the wording of the region required to decide what constructions are needed.**

Breaking down the wording of the question we have the following three facts,

- closer to the side AD than the side AB,
- more than 2 \ m from A,
- less than 3 \ m from the side AB.

To determine the points closer to the side AS than the side AB, we need to construct an angle bisector. This is because all of the points that are on the angle bisector are the same distance from both lines, and so points in one half will be closer to one line than the other.

To determine the points more than 2 \ m from A, we need to construct a circle with radius 2 \ cm. This is because A is a point and we need all the points beyond 2 \ m from A.

To determine the points less than 3 \ m from the side AB, we need to construct a parallel line that is 3 \ cm from the line AB.

**Perform any relevant constructions for points or line segments involved.**

**Indicate the region required as necessary.**

The shaded area is the locus of points which satisfy all of Ellie’s criteria,

- closer to the side AD than the side AB,
- more than 2 \ m from A,
- less than 3 \ m from the side AB.

The bearing of a ship from lighthouse A is 130^{\circ}. The bearing of the same ship from lighthouse B is 025^{\circ}. Mark the position of the ship on the diagram.

**Locate the point you are measuring the bearing from and draw a north line if there is not already one given.**

We need to draw bearings from both points A and B so both need a north line.

**Draw a line from the start point in the direction of the bearing. If you are producing a scale drawing and know the distance to locate a point use this scale appropriately.**

The ship is located at the point where the two lines intersect.

A plane starts at airport A. The plane flies 40 \ km east and then 60 \ km north before landing at airport B. Find the bearing of airport B from airport A.

**Locate the points you are calculating the bearing from and to.**

Drawing a diagram is very helpful here.

**Using the north lines for reference at both points, use angle rules and/or trigonometry to calculate any angles that are required.**

We can find angle a using SOHCAHTOA and then use angle facts to determine angle x.

\begin{aligned} \tan (a) &=\frac{60}{40}\\\\ a&=\tan ^{-1} (\frac{60}{40})\\\\ a&=56^{o} \end{aligned}

As the sum of angles in a right angle is 90 degrees, we can find the missing angle x by subtracting 56 from 90 to get 90-56=34^{\circ}

**Read off the three-figure bearing required.**

Converting the angle of 34^{\circ} to a three-figure bearing gives us the solution 034^{\circ}.

**Step-by-step guide:** Constructing triangles

**See also:** How to draw a hexagon

**Removing construction lines**

When drawing constructions, the construction lines must be shown and not erased.

**Corners when drawing the locus around a polygon or line**

When drawing the locus around a polygon or a line, the locus around a vertex should be an arc. For example,

**Connecting the corners instead of bisecting an angle**

When finding the locus of points equidistant from two sides of a rectangle, we need to bisect the angle rather than drawing a line between the two corners.

For example,

**Not measuring bearings from north**

It is important to remember that bearings are always measured from north.

**Not measuring bearings in a clockwise direction**

Bearings must be measured in a clockwise direction.

For example, the bearing of A from B is 360-100=260^{\circ}, not 100^{\circ}.

1. What type of construction is shown below?

Angle bisector of the angle AB

Perpendicular to the line AB at the point P

Perpendicular bisector of the line AB

Triangle ABC

The two straight lines are at right angles to each other and so they are perpendicular lines. Also, the radius of each arc from A and B is the same and so the line is bisected into two equal halves.

2. Which diagram correctly shows how to construct the angle bisector of angle ABC?

An arc of the same radius is drawn on line AB and BC from the point B. From these arcs, two intersecting arcs are drawn with a straight line connecting the vertex at B and the intersection point.

3. Which diagram shows the locus of points a distance of d from the triangle?

Each vertex should be treated as a point and therefore the locus around each vertex should be an arc. The locus of points from a line are parallel.

4. Hannah wants to shade the area closer to side AB than side AD, and more than 2 \ cm from point B. Which diagram shows the correct area?

We need to draw a circle with radius 2 \ cm around the point B and the angle bisector of angle BAD. The shaded region is outside of the circle and to the left side of the line drawn for the angle bisector.

5. Find the bearing of P from Q.

065^{\circ}

115^{\circ}

295^{\circ}

025^{\circ}

Bearings are measured clockwise from north. The bearing of P from Q is 360-65=295^{\circ}.

6. The bearing of airport B from airport A is 100^{\circ}. What is the bearing of airport A from airport B?

260^{\circ}

280^{\circ}

80^{\circ}

190^{\circ}

By drawing a diagram, we can see that the co-interior angle at B is 180-100=80^{\circ} (remember that all north lines are parallel). Therefore the bearing of A from B, measured clockwise from north, is 360-80=280^{\circ}.

1. Construct the perpendicular to the line AB at the point P.

**(3 marks)**

Show answer

Two arcs equidistant from P.

**(1)**

Two further arcs above (or below) P.

**(1)**

Perpendicular drawn and all construction lines shown.

**(1)**

2. Shade the area that is closer to the side AB than the side BC **and** is closer to the point B than the point A.

**(3 marks)**

Show answer

Perpendicular bisector of AB.

**(1)**

Angle bisector of angle ABC.

**(1)**

Correct area shaded.

**(1)**

3. Some rangers are tracking the position of a lion.

The lion is on a bearing of 055^{\circ} from ranger A and 290^{\circ} from ranger B.

Mark the position of the lion on the diagram below.

**(3 marks)**

Show answer

Bearing of 055^{\circ} from A drawn.

**(1)**

Bearing of 290^{\circ} from B drawn.

**(1)**

Position of lion marked.

**(1)**

You have now learned how to:

- Derive and use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle); recognise and use the perpendicular distance from a point to a line as the shortest distance to the line
- Interpret and use bearings

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