Constructions, Loci and Bearings

This lesson covers ruler-and-compass constructions, loci (the set of points satisfying a given condition), and three-figure bearings. These are practical geometry skills tested in Edexcel GCSE Mathematics, often requiring you to draw accurately with a compass and ruler.

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Key Vocabulary

Term	Meaning
Construction	A geometric drawing made using only a compass and straight edge (ruler without measurements)
Locus (plural: loci)	The set of all points that satisfy a given condition
Bearing	A direction measured clockwise from north, given as a three-figure number (e.g. 045°, 210°)
Perpendicular bisector	A line at right angles to a line segment that passes through its midpoint
Angle bisector	A line that divides an angle exactly in half
Equidistant	The same distance away

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Standard Constructions

You must be able to perform these constructions using only a pair of compasses and a ruler (straight edge). Construction marks (arcs) must be shown — do not erase them.

1. Perpendicular Bisector of a Line Segment

This construction finds the midpoint and draws a line at right angles.

Method:

1. Open the compass to more than half the length of the line segment AB.
2. Place the compass point at A and draw arcs above and below the line.
3. Without changing the compass width, place the point at B and draw arcs above and below, crossing the first arcs.
4. Draw a straight line through the two intersection points.

This line is the perpendicular bisector. Every point on it is equidistant from A and B.

2. Angle Bisector

This construction divides an angle exactly in half.

Method:

1. Place the compass point at the vertex of the angle and draw an arc crossing both arms.
2. Place the compass point at each intersection and draw two arcs that cross each other inside the angle.
3. Draw a straight line from the vertex through the intersection of the arcs.

3. Perpendicular from a Point to a Line

Method:

1. Place the compass at the point and draw an arc crossing the line at two places.
2. From each crossing point, draw arcs below the line that intersect.
3. Draw a line from the original point through the intersection.

4. Perpendicular from a Point on a Line

Method:

1. Place the compass at the point on the line and draw arcs equal distances either side.
2. From each arc, draw arcs above (or below) the line that intersect.
3. Draw a line from the original point through the intersection.

5. Constructing an Equilateral Triangle

Method:

1. Draw a base line AB of the required length.
2. Set the compass to the length AB.
3. Place the point at A and draw an arc above the line.
4. Place the point at B and draw an arc crossing the first.
5. The intersection is the third vertex C. Join AC and BC.

6. Constructing a 60° Angle

Follow the same method as constructing an equilateral triangle — the angle at A (or B) is 60°.

7. Constructing a 90° Angle

Construct the perpendicular bisector of a line segment — the resulting angle is 90°.

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Loci

A locus is the set of all points that satisfy a particular condition. In GCSE questions, you typically shade a region or draw a line/curve.

Four Standard Loci

Condition	Locus
Fixed distance from a point	Circle (radius = given distance)
Fixed distance from a line	Two parallel lines either side, with semicircles at each end
Equidistant from two points	Perpendicular bisector of the line joining the two points
Equidistant from two lines that meet at a point	Angle bisector of the angle between the two lines

Worked Example 1

A treasure is buried within 5 m of a tree T and closer to fence AB than fence CD (where AB and CD meet at a point). Shade the region where the treasure could be.

Step 1: Draw a circle of radius 5 m (to scale) centred on T. Step 2: Construct the angle bisector of the angle between AB and CD. Step 3: Shade the region that is inside the circle AND on the AB side of the bisector.

Worked Example 2

A goat is tied to a post P by a 6 m rope. Shade the region the goat can reach.

The goat can reach all points within 6 m of P — so the locus is a circle of radius 6 m centred on P.

Worked Example 3

Find the locus of points equidistant from A and B, where A and B are 8 cm apart.

Construct the perpendicular bisector of AB. Every point on this line is the same distance from A as from B.

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Combining Loci

Exam questions often combine two or more conditions. The answer is the region where all conditions are satisfied simultaneously (the intersection).

Worked Example 4

In a rectangular garden ABCD (12 m by 8 m), a tree must be:

• More than 3 m from the edge AB
• Within 5 m of corner C

Draw a line parallel to AB, 3 m away (inside the garden). Draw an arc of radius 5 m centred on C. The region satisfying both conditions is where these overlap.

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Bearings

A bearing is a direction measured clockwise from north. Bearings are always written as three figures.

Direction	Bearing
North	000°
East	090°
South	180°
West	270°
North-East	045°
South-West	225°

Rules for Bearings

1. Always measure from north.
2. Always measure clockwise.
3. Always give three figures (e.g. 045°, not 45°).

Worked Example 5

The bearing of B from A is 065°. Find the bearing of A from B.

The bearing of A from B = 065° + 180° = 245°

Rule: If the bearing from A to B is x°, the bearing from B to A is x° + 180° (if x < 180°) or x° - 180° (if x >= 180°).

Worked Example 6

A ship sails from P on a bearing of 130° for 50 km to Q, then on a bearing of 230° for 40 km to R. Find angle PQR.

At Q, the angle between the direction from P (reverse bearing = 130° + 180° = 310°) and the direction to R (230°) is: 310° - 230° = 80°

Worked Example 7

Town B is due east of town A. Town C is on a bearing of 140° from A. Find angle BAC.

Bearing of B from A = 090° (due east). Bearing of C from A = 140°. Angle BAC = 140° - 90° = 50°

Bearings and Scale Drawings

Many bearing questions require a scale drawing. Use a protractor to measure angles from north and a ruler to measure distances to scale.

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Common Mistakes and Misconceptions

• Erasing construction arcs. Keep them — the examiner needs to see them to award marks.
• Not using a compass. Constructions must be done with a compass, not estimated by eye.
• Measuring bearings from south or anticlockwise. Always from north, always clockwise.
• Writing bearings as two figures. 45° must be written as 045°.
• Confusing the perpendicular bisector with the angle bisector. Perpendicular bisector = equidistant from two points; angle bisector = equidistant from two lines.
• Not drawing the locus for a distance from a line correctly. Remember to include the semicircular ends.

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Edexcel Exam Tips

• Bring a sharp pencil, compass and protractor to the exam.
• For construction questions, make your arcs clear and do NOT erase them.
• For loci questions, shade the required region clearly and show all construction lines.
• Bearing questions often combine with trigonometry or Pythagoras — read carefully.
• If a bearing question involves a scale drawing, state the scale you are using.
• On the Edexcel paper, constructions/loci questions are typically worth 3-4 marks.

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Practice Problems