- Describe the locus of points that are exactly 3 cm from a fixed point .[2]
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A phone mast must be within 3 km of a village but closer to road A than road B. Where can it go? Questions like this are answered with a ruler and a pair of compasses. A is the set of all points that obey a rule, and the constructions in this lesson draw loci accurately.
The big picture
Four constructions do almost all of the work: the perpendicular bisector of a line, the bisector of an angle, the perpendicular from a point to a line, and the perpendicular at a point on a line. Each one is built from arcs of equal radius.
Each standard locus matches a construction. Equidistant from two points is a perpendicular bisector; equidistant from two lines is an angle bisector. Combining loci and shading the region that satisfies every condition is the typical exam question.
What you'll learn
Set the compasses to more than half the length of .
Draw arcs above and below the line from , then from with the same radius.
Join the two crossing points. This line cuts in half at .
What is the name of the line containing every point that is the same distance from two fixed points and ?
Every point on the perpendicular bisector is the same distance from as from . That is why it is also the locus of points equidistant from two points.
Angle bisector: from the vertex, draw an arc cutting both arms. From each crossing point, draw arcs of equal radius that meet. Join the vertex to that point.
Perpendicular from a point to a line: draw an arc from cutting the line twice, then construct the perpendicular bisector of those two points.
An angle of comes from an equilateral triangle (same radius throughout); bisecting it gives .
Tip — Leave all construction arcs visible. Rubbing them out removes the evidence the marks depend on.
A fixed distance from a point: a circle.
A fixed distance from a line segment: two parallel lines joined by semicircles at the ends.
Equidistant from two points: the perpendicular bisector.
Equidistant from two intersecting lines: the angle bisector.
A goat is tied to a post in the middle of a large field by a rope 5 m long. What area can it graze, to 1 decimal place?
Tip — Check whether a boundary is included: "less than" gives a dashed boundary, "at most" a solid one.
Think like an examiner
Remember these
Watch out for these
Stretch yourself
A goat is tied to one corner of a rectangular shed measuring 4 m by 3 m, with a rope 5 m long. The goat stays outside the shed. Find the area it can graze, in terms of and to 3 significant figures.
Hint — Sketch the locus: a large circular sector, plus smaller sectors where the rope bends around the shed’s corners.
Questions students ask
Key takeaways
Exam-style questions
3 original questions · 8 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.
A rectangular field is 40 m by 25 m, with m. A sprinkler at the midpoint of waters every point of the field within 12 m of it.
Triangle is drawn accurately.
Test yourself
Ready to practise Constructions & Loci? Pick a mode and earn XP & Dobloons.
Real past-paper questions on Constructions & Loci, marked mark-by-mark. How you do feeds straight into your weak-topic list, so your revision keeps targeting what actually needs work.