Constructions, Loci and BearingsAQA GCSE Maths: Revision notes
Section 1
How do you construct standard geometric shapes using ruler and compass?
Ruler and compass constructions are exact geometric drawings used to create precise shapes and lines without measurement.
Key constructions you must master:
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Constructing an equilateral triangle
- Draw a line segment AB (the base)
- Set compass to length AB
- Draw an arc from point A above the line
- Draw an arc from point B to intersect the first arc at point C
- Join AC and BC
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Constructing the perpendicular bisector of a line segment
- Set compass to more than half the length of the line segment
- Draw arcs from each endpoint, above and below the line
- Draw arcs from the other endpoint to intersect the first arcs at two points
- Draw a straight line through these two intersection points
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Constructing the angle bisector
- Draw an arc from the angle's vertex, intersecting both arms
- From each intersection point, draw equal arcs that meet inside the angle
- Draw a line from the vertex through this intersection point
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Constructing a perpendicular from a point to a line
- If the point is on the line: use the perpendicular bisector method
- If the point is off the line: draw arcs from the point to intersect the line at two points, then construct the perpendicular bisector of that segment
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Constructing angles of 60°, 30° and 90°
- 60°: construct an equilateral triangle
- 30°: bisect a 60° angle
- 90°: construct a perpendicular
Always show all compass arcs and construction lines on your diagram — examiners want to see how you arrived at your answer, not just the final shape. Leave arcs visible; do not erase them.
Students often fail to set the compass to exactly the right width. For a perpendicular bisector, the compass must be wider than half the line segment, and for angle bisectors, the compass width must remain constant when drawing the two intersection arcs.
Section 2
What is a locus and how do you use loci to solve problems?
A locus (plural loci) is the set of all points that satisfy a given condition or rule. Loci problems often involve finding regions where multiple conditions overlap.
Standard loci you need to recognise:
| Condition | Locus |
|---|---|
| Points at a fixed distance from a point | A circle |
| Points at a fixed distance from a line | Two parallel lines (one on each side) |
| Points equidistant from two fixed points | The perpendicular bisector of the line joining them |
| Points equidistant from two intersecting lines | The two angle bisectors |
| Points closer to point A than point B | The region on the A side of the perpendicular bisector |
How to solve locus problems:
- Identify each condition separately
- Draw the locus for each condition (circle, parallel lines, angle bisectors, etc.)
- Shade or identify the region where all conditions are satisfied simultaneously
- If asked for specific points, find intersections of loci or points within the region
Common applications:
- Finding areas suitable for buildings (within distance of road AND away from hazards)
- Locating radar coverage zones (distance from transmitter)
- Navigation and territorial boundaries
- Garden or field design problems
A point P is equidistant from two towns A and B, and less than 10 km from town A. The locus of points equidistant from A and B is the perpendicular bisector of line AB. The locus of points less than 10 km from A is the interior of a circle radius 10 km centred at A. The solution region is the part of the perpendicular bisector that lies inside the circle.
When a locus question asks for a 'region', shade or clearly indicate the area. If it asks for points on a locus, mark them with dots or crosses, not shaded areas.
Section 3
How do three-figure bearings work and how are they used?
A bearing is a direction expressed as an angle measured clockwise from north. Three-figure bearings are written as three digits ranging from 000° to 360°.
Key facts about bearings:
- North is always 000° (or 360°)
- East is 090°
- South is 180°
- West is 270°
- All bearings are measured clockwise from north
- Bearings are always written with three digits (e.g. 045°, not 45°)
Converting between bearings and compass directions:
- NE (northeast) = 045°
- SE (southeast) = 135°
- SW (southwest) = 225°
- NW (northwest) = 315°
Solving bearing problems:
- Draw a diagram with a north line at each relevant point
- Mark all bearings as angles measured clockwise from north
- Use scale drawings or trigonometry to find distances or positions
- Remember: bearings from A to B are not the same as bearings from B to A (they differ by 180°)
The back bearing rule: If the bearing from A to B is θ, the bearing from B to A is (θ + 180°) or (θ - 180°) depending on whether you go over or under 360°
Common problem types:
- Finding position after travel on multiple bearings
- Determining bearings between known locations
- Calculating distances using scale drawings with bearing diagrams
A ship travels from port A on a bearing of 120° for 50 km. On a scale drawing where 1 cm = 10 km, draw a north line at A, measure 120° clockwise, and mark a point 5 cm along this line. To find the bearing from the destination back to A, add 180° to 120° = 300°.
Students often forget to add 180° to the forward bearing to get the back bearing, or they subtract when they should add (especially when the result would exceed 360°). Always use the rule: back bearing = forward bearing ± 180° (adjusting to stay between 000° and 360°).
Always draw a north line at EACH point where bearings are involved, and measure bearings clockwise from this line. This prevents confusion and helps you visualise the problem correctly.
Section 4
How do scale factors and scale drawings work?
A scale drawing is a reduced or enlarged representation of a real object or area. The scale factor determines the relationship between measurements on the drawing and real-world measurements.
Understanding scale notation:
- 1 : n means 1 unit on the drawing represents n units in reality
- 1 cm : d km means 1 cm on the drawing represents d km in real life
- Scale factor = real distance ÷ drawn distance
Using scale to convert distances:
- Measure the distance on the drawing (in the units given in the scale)
- Multiply by the scale factor to find the real distance
- OR divide the real distance by the scale factor to find the drawn distance
Example scale conversions:
| Scale | Meaning |
|---|---|
| 1 : 100 | 1 unit on drawing = 100 units in reality |
| 1 : 50,000 | 1 cm on drawing = 50,000 cm (500 m) in reality |
| 1 : 1,000,000 | 1 cm on drawing = 10 km in reality |
Working with bearings and scale drawings together:
- Draw a diagram to scale showing positions of locations
- Use a protractor to measure bearings (angles from north)
- Measure distances on the drawing with a ruler
- Convert measured distances back to real distances using the scale
Important: Areas and volumes scale differently
- Linear distances scale by the scale factor
- Areas scale by the square of the scale factor
- Volumes scale by the cube of the scale factor
- If the linear scale factor is k, the area scale factor is k²
A map has a scale of 1 : 25,000. Two towns are 8 cm apart on the map. Real distance = 8 cm × 25,000 = 200,000 cm = 2 km. Conversely, if towns are 5 km apart in reality, on the map they are 5 km ÷ 25,000 = 500,000 cm ÷ 25,000 = 20 cm apart.
When working with scales, always ensure your measurements and scale are in compatible units. Convert km to cm or vice versa before calculating. Marks are often lost through unit confusion rather than method errors.
Think of a scale drawing like a photocopy — if you photocopy at 50% scale, all distances become half size. If you photocopy at 200%, all distances double. The scale factor works exactly the same way.
Section 5
How do you apply constructions, loci and bearings to solve real problems?
Examination questions often combine multiple concepts. Here's how to approach integrated problems:
Problem-solving strategy:
- Read carefully and identify all constraints (distances, angles, conditions)
- Decide which tools to use: constructions (for exact angles/bisectors), loci (for regions), bearings (for directions), scale drawings (for distance problems)
- Draw and label all necessary elements on a diagram
- Work systematically through each condition
- Give final answers in the form requested (a region, a point, a bearing, a distance)
Common question types:
- "Find the region where..." → Use loci; shade the intersection of all conditions
- "Construct..." → Use ruler and compass; show all arcs and construction lines
- "Find the bearing from X to Y" → Draw north lines, use protractor, give answer as three-figure bearing
- "How far is it really?" → Use scale factor to convert measured distance
- "Plot the course..." → Combine bearings with scale drawings; measure results
Quality of answers:
- Constructions must be accurate and show all working
- Loci diagrams must clearly show the region (shading, hatching or annotation)
- Bearings must be marked as three-figure values from north lines
- Scale work must show the conversion calculation explicitly
- Always state units (km, m, degrees, etc.)
Checking your work:
- Does your diagram match all stated conditions?
- Are all measurements consistent with the scale?
- Have you answered the exact question asked?
- Are bearings between 000° and 360°?
- Is your shading or region identification clear?
A question might state: 'A campsite must be more than 500 m from a road (drawn as line AB) and equidistant from two towns C and D. Scale 1 cm = 100 m. Show the possible region.' Draw the perpendicular bisector (locus for equidistant from C and D). Draw parallel lines 500 m away from the road (use scale: 5 cm from line AB). Shade the intersection. Show all construction lines.
In multi-part problems, work in the order requested and don't skip diagram elements. Examiners give marks for showing the perpendicular bisector even if your final region is slightly wrong — the method is being assessed.
Must Know
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Constructions require a ruler and compass; always show all arcs and construction lines on your diagram. Perpendicular bisectors and angle bisectors are constructed using equal arc methods.
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A locus is the set of all points satisfying a condition. Key loci are circles (fixed distance from a point), parallel lines (fixed distance from a line), perpendicular bisectors (equidistant from two points), and angle bisectors (equidistant from two lines). Multiple loci problems require you to find the intersection or overlap region.
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Three-figure bearings are always measured clockwise from north and written with three digits (000° to 360°). The back bearing differs by 180°. North = 000°, East = 090°, South = 180°, West = 270°.
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Scale drawings use a scale factor to convert between drawn and real distances. A scale of 1:n means 1 unit on the drawing represents n units in reality. Always convert to compatible units before calculating. Areas scale by the square of the linear scale factor; volumes scale by the cube.
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Integrated problems combine constructions, loci, bearings and scale. Read all constraints, draw systematically, label clearly, and answer in the exact form requested (region, point, bearing, distance with units).
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Accuracy and presentation matter: show all working, leave construction lines visible, use correct notation (three-figure bearings, scales), and ensure diagrams are to scale where specified.
That's the notes covered.
Carry on to the next subtopic.