Loci (Edexcel GCSE Maths): Revision Notes
Loci
What is a locus?
A locus is a set of points that all satisfy the same condition or rule. The plural of locus is loci. The concept of loci is fundamental in geometry and helps us understand how points relate to each other based on specific geometric conditions.
You can construct loci using a ruler and pair of compasses. Sometimes the locus forms a line or curve, but it can also be a region containing many points. This makes loci particularly useful for solving real-world problems involving distances and positions.
Types of loci
Understanding the different types of loci helps you recognise patterns and apply the correct construction methods for various geometric problems.
Circle locus
When you want to find all points that are the same distance from a fixed point, you get a circle.
Worked Example: Circle Locus Construction
Problem: Find all points that are exactly 7 cm away from point A.
Solution: All points that are exactly 7 cm away from point A will form a circle with centre A and radius 7 cm. Points that are less than 7 cm from A will lie inside this circle, while points further than 7 cm will lie outside the circle.
Perpendicular bisector
When you want to find all points that are equidistant from two fixed points, you get the perpendicular bisector of the line joining those two points.
Worked Example: Perpendicular Bisector
Problem: Find all points that are the same distance from point B as they are from point C.
Solution: All points that are the same distance from point B as they are from point C will lie on the perpendicular bisector of line BC. This line cuts BC in half at a right angle, ensuring every point on it is equidistant from both B and C.
Parallel line locus
When you want to find all points that are the same distance from a straight line, you get two parallel lines on either side of the original line, plus semicircles at each end.
For example, all points that are exactly 2 cm away from line ST will form two parallel lines 2 cm away on each side, connected by semicircles at the ends.
Remember: The shortest distance from any point to a line is always the perpendicular distance. This is crucial for accurate locus construction.
Combining conditions
Sometimes you need to find regions that satisfy more than one condition at the same time. This requires careful analysis and systematic construction of each individual locus.
Worked Example: Multiple Conditions
Problem: Shade the region that is:
- More than 6 cm from point D, AND
- Closer to line BC than to line AD
Solution:
- First, construct a circle with centre D and radius 6 cm (points more than 6 cm from D lie outside this circle)
- Then, construct the perpendicular bisector of the lines BC and AD (points closer to BC lie on one side of this bisector)
- Finally, shade the region where both conditions are satisfied simultaneously
In these cases, you construct each locus separately, then find where they overlap to satisfy both conditions.
Scale drawing and construction
Accurate construction is essential for solving loci problems effectively. The precision of your geometric constructions directly affects the accuracy of your solutions.
Construction Guidelines:
- Use a compass to draw circles and arcs
- Set your compass to the correct measurement on your ruler
- Place the compass point exactly on the required position
- Keep the compass setting fixed while drawing
- Use a ruler for straight lines and perpendicular bisectors
Exam tip: In scale drawing questions, always check what distance 1 cm represents in real life, then adjust your compass accordingly.
Worked example applications
Loci are used in real-world problems like finding safe zones, coverage areas, or regions with specific properties. Understanding these applications helps you see the practical value of geometric loci.
Real-World Application: Lifeguard Tower
Problem: A lifeguard tower can monitor swimmers within 30 m. What is the safe swimming area?
Solution: The safe swimming area would be a circle with radius 30 m (to scale) centred on the tower. Any swimmer within this circular region can be effectively monitored by the lifeguard.
When answering loci questions, follow this systematic approach:
- Read the conditions carefully and identify what type of locus each represents
- Identify which type of locus each condition creates
- Construct each locus accurately with compass and ruler
- Shade or mark the final region clearly to show your answer
Key Points to Remember:
- A locus is a set of points satisfying a condition (plural: loci)
- Fixed distance from a point = circle
- Same distance from two points = perpendicular bisector
- Fixed distance from a line = parallel lines with semicircular ends
- Use compass and ruler for accurate construction
- Combine conditions by finding overlapping regions