Loci drawn and checked on a grid, regions that follow two rules, where two loci meet, map scales and bearings for a journey.
What you will learn
- Draw the locus of points a fixed distance from a point or a line
- Use the angle bisector as the locus equidistant from two lines
- Find where two loci meet and count the points in a region
- Use a map scale and bearings to find a position and a distance
The four loci
- Points a fixed distance from a point: a circle.
- Points a fixed distance from a long line: two parallel lines, one each side.
- Points the same distance from two points: the perpendicular bisector.
- Points the same distance from two lines that meet: the angle bisector.
"Less than" or "within" is a region: inside the circle, or between the lines. "Closer to A than to B" is the side of the perpendicular bisector that A is on.
To plot points of a circle of radius r about P on a grid, use Pythagoras: a point dx across from P is h above it, where dx² + h² = r². For r = 5, a point 3 across is √(25 − 9) = 4 above P. The whole locus is a circle, centre P, radius r (not the diameter), and the region inside it has area π × r².
For a region given by two loci, a point must pass both tests. To count whole-number points in it, test each point: is it on A's side of the bisector (closer to A, so its distance squared to A is less than to B), and is its distance squared from C at most r²?
The rest of this lesson and its 11 practice questions are locked.
In this lesson
- The four loci
- Distances at right angles (Higher)
- Where loci meet
- Scales and bearings
11 practice questions, marked as you go; about 25 minutes in all.
Key terms
Two lines
In this unit: Drawing and constructing
The answers papers ask you to draw: graphs plotted from a table, sketches with their key points, statistical charts, ruler-and-compass constructions and loci, each checked the way a mark scheme checks it.
- Plotting graphs
- Sketching graphs
- Drawing charts
- Constructions on a grid
- Loci and bearings