OCR · GCSE · Mathematics
Constructions
Master OCR GCSE Constructions by learning to use a ruler and compasses with precision. This guide breaks down perpendicular bisectors, angle bisectors, and complex loci to help you secure every mark. Forget the protractor and learn the techniques that examiners reward.
- 4 min read
- 3 worked examples
- 5 practice questions
- 6 key terms
Study Notes
Overview
Constructions is a topic that rewards precision and method. In your OCR GCSE Mathematics exam, you will be asked to create accurate geometric figures using only a pair of compasses and a ruler. This is a test of your ability to follow a process, not just get the right answer. Examiners are looking for clear evidence of your method, which means leaving all your construction arcs visible. This guide will cover the core constructions required for both Foundation and Higher tiers, including perpendicular bisectors, angle bisectors, and the more complex topic of loci. Understanding constructions is not just about drawing lines; it connects to deeper geometric principles and has applications in fields like architecture, engineering, and design. Typical exam questions will command you to 'Construct' a figure, and may involve shading regions that satisfy multiple conditions, especially at the Higher tier.
Key Concepts
Concept 1: Perpendicular Bisector
A perpendicular bisector is a line that cuts another line segment into two equal halves at a 90-degree angle. Think of it as the ultimate line of symmetry for a segment. To construct it, you set your compass to a radius that is more than half the length of the line segment. From each endpoint of the segment, you draw an arc above and below the line. The points where these arcs intersect are the key. Drawing a straight line between these intersection points gives you the perpendicular bisector. This construction is fundamental because it also represents the locus of all points that are equidistant from the two endpoints of the segment.
Example: Given a line segment AB of 8cm, the perpendicular bisector will pass through the midpoint at 4cm and be at a right angle to AB.
Concept 2: Angle Bisector
An angle bisector is a line that divides an angle into two smaller, equal angles. It's like finding the exact middle of an angle. To construct it, you place your compass on the vertex of the angle and draw an arc that crosses both arms of the angle. From each of these intersection points, you draw another pair of arcs in the middle of the angle. Where these arcs cross, you draw a line from the vertex through this intersection point. This new line is your angle bisector. This is crucial for problems involving loci equidistant from two lines.
Example: Bisecting a 60-degree angle results in two 30-degree angles.
Concept 3: Loci and Regions
A locus is a path or set of points that satisfies a certain rule. For example, the locus of points a fixed distance from a point is a circle. The locus of points equidistant from two points is a perpendicular bisector. Higher-tier questions often combine multiple loci to define a specific region. You might be asked to shade the area that is 'closer to point A than point B' AND 'less than 5cm from point C'. This requires you to construct a perpendicular bisector and a circle, and then identify the correct region that satisfies both conditions. These questions test your ability to translate written conditions into geometric constructions.
Example: Shading the region inside a rectangle that is closer to side AD than side BC.
Mathematical/Scientific Relationships
- Perpendicular Bisector Theorem: Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.
- Angle Bisector Theorem: Any point on the bisector of an angle is equidistant from the two sides of theangle.
Practical Applications
- Architecture and Engineering: Constructions are used to create accurate blueprints and plans for buildings, bridges, and other structures.
- Navigation: Loci are used in GPS systems to pinpoint locations based on distances from satellites.
- Design: Graphic designers use construction principles to create symmetrical and balanced logos and layouts.
Visual Resources
3 diagrams and illustrations
Interactive Diagrams
2 interactive diagrams to visualise key concepts
Conceptual Flow Outline
Flowchart showing the steps to construct a perpendicular bisector.
Conceptual Flow Outline
Flowchart showing the steps to construct an angle bisector.
Worked Examples
3 worked examples — open one to explore the question and available guidance.
Practice Questions
Test your understanding — click to reveal model answers
Draw a line segment 7cm long. Construct the locus of points that are 3cm from the line segment.
Hint: Think about the shape you would get if you traced all the points 3cm away from the line. What happens at the ends?
Construct a triangle ABC with AB = 8cm, AC = 6cm and BC = 7cm. Construct the bisector of angle BAC. Show all your construction lines.
Hint: First, construct the triangle accurately using a ruler and compasses. Then, perform the angle bisector construction at vertex A.
Two radio transmitters are located at points S and T, which are 10km apart. The transmitter at S has a range of 6km and the transmitter at T has a range of 8km. A walker is hiking on a path that is always equidistant from S and T. Show, by construction, the part of the path where the walker can receive a signal from both transmitters. Use a scale of 1cm to 1km.
Hint: The path is the perpendicular bisector. The ranges of the transmitters are circles. Where do these three loci overlap?
Construct an equilateral triangle of side length 5cm.
Hint: All sides are equal, and all angles are 60°. You can use this to your advantage.
A point P is equidistant from the lines AB and AC. Describe the locus of P.
Hint: What construction creates points that are the same distance from two intersecting lines?