All the points that fit
Locus of points
A rule for points
Mark a point O on a sheet of paper. Now mark every point on the sheet that is exactly 5 units away from O, not one point more and not one less.
What do you think all those points make together?
What this lesson covers
The idea
The set of points that satisfy a given condition is called the locus of points satisfying that condition; a circle is the locus of points equidistant from a given point.
A rule for points
Mark a point O on a sheet of paper. Now mark every point on the sheet that is exactly 5 units away from O, not one point more and not one less.
What do you think all those points make together?
- A circle
- A straight line
- A whole round patch, filled in
Test the points
Some points are marked around O. Test them one by one: how far is each from O?
The locus
The set of points that satisfy a given condition is called the locus of points satisfying that condition. A circle is the locus of points equidistant from a given point.
Here the condition was "5 units from O". The points that fit it make a circle. Points that do not fit are not on it.
Notes
The set of points that satisfy a given condition is the locus of those points. A circle is the locus of points equidistant from a given point.
Check yourself
What is the locus of all points that are 3 cm from a point O?
The point O is the origin. Which of these points is 5 units from O?
The locus of points 10 units from O is a circle. How far from O is the point (6, 8)? Give the distance in units.
Answer: 10 units
√(6² + 8²) = √(36 + 64) = √100 = 10. So (6, 8) is on that circle.
What does "the locus of points satisfying a condition" mean?
- A filled disc: the circle and everything inside it. Points inside the circle are less than 3 cm from O, so they do not satisfy "exactly 3 cm".
- A circle with centre O and radius 3 cm — correct. Yes. The points at exactly 3 cm from O make that circle.
- A straight line 3 cm long. A line segment has points at many different distances from O. Only a circle keeps the distance the same.
- (4, 4). √(4² + 4²) = √32, about 5.7. That is more than 5.
- (2, 3). √(2² + 3²) = √13, about 3.6. That is less than 5.
- (3, 4) — correct. Yes. √(3² + 4²) = √25 = 5.
- (5, 5). √(5² + 5²) = √50, about 7.1. That is more than 5.
- The set of all points that satisfy the condition — correct. Yes. A circle is the locus of the points that are the same distance from a given point.
- The centre of a circle. The centre is one point. A locus is a whole set of points.
- The longest distance between two points. A locus is not a distance. It is the set of points that follow a rule.