Equal distance from two points
Perpendicular bisector.
Learning Objectives
- construct perpendicular and angle bisectors;
- construct 60ยฐ, 90ยฐ and common derived angles;
- construct triangles accurately;
- understand basic loci;
- use ruler and compasses correctly.
Introduction
This lesson develops Constructions and Loci from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
Construction Rules
Keep compass arcs visible. They are part of the mathematical evidence.
Perpendicular Bisector
- Open compasses to more than half the segment length.
- Draw arcs above and below from each endpoint.
- Join the arc intersections.
Angle Bisector
- Draw an arc from the vertex crossing both arms.
- From the two crossing points, draw equal arcs that meet.
- Join the vertex to the new intersection.
Basic Loci
Equal distance from two intersecting lines
Angle bisectors.
Fixed distance from a point
Circle.
Fixed distance from a line
Two parallel lines.
Try It: Explore the Main Constructions
๐ง Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
Worked Examples
Reason through the method
- Identify the known information and what must be found.
- Choose the definition, property or formula that connects them.
- Substitute carefully and show each step.
- Check the notation, units and whether the result is sensible.
Practice Questions
- Describe how to construct the perpendicular bisector of AB.
- Which locus contains points equidistant from A and B?
- Which locus contains points 4 cm from point P?
- What angle is formed when a perpendicular is constructed?
- Why should construction arcs remain visible?
โ Check Your Work
Show answers and working
- Draw equal-radius arcs from A and B above and below the segment, then join their intersections.
- The perpendicular bisector of AB.
- A circle centred at P with radius 4 cm.
- 90ยฐ.
- They show the valid compass construction and method.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Use ruler and compasses, not estimation.
- Bisectors create equal parts.
- Loci describe sets of positions satisfying a rule.