Constructions and loci
Maths · GCSE · The School
Construct means compasses
In this topic "construct" has a narrow meaning: produce the figure using compasses and a straight edge, with the arcs still showing. Measuring an angle with a protractor and drawing the line scores nothing, however accurate it is, because the marks are for the method and the arcs are the evidence of it. Set the compasses, do not change the radius halfway through unless the method says to, and never rub the arcs out. A construction with no visible arcs looks exactly like a guess.
The perpendicular bisector
To bisect a line AB: open the compasses to more than half of AB, draw arcs above and below from A, repeat from B without changing the radius, and join the two crossing points. The line you get cuts AB in half at 90 degrees. It is worth knowing what that line IS rather than just how to draw it: it is the set of all points the same distance from A as from B. That is why it answers questions about being equally far from two towns, two trees or two posts.
The angle bisector
To bisect an angle: put the compass point on the vertex and draw an arc crossing both arms. From each crossing point draw an arc into the middle, and join the vertex to where those two arcs meet. The line splits the angle in half. Again, the meaning matters: it is the set of points the same distance from one arm as from the other. Equidistant from two POINTS gives a perpendicular bisector; equidistant from two LINES gives an angle bisector, and mixing those two up is the usual reason a locus answer comes out the wrong shape.
A locus is just the set of points obeying a rule. Fixed distance from a point is a circle. Fixed distance from a straight line is a pair of parallel lines. Fixed distance from a line SEGMENT is a racetrack: two parallel lines joined by semicircular ends. A goat is tied to a post by a 5 m rope and must stay at least 3 m from a wall. Draw a circle of radius 5 about the post, draw a line 3 m from the wall on the near side, and shade only where inside the circle meets beyond that line. Two conditions means two boundaries and one overlap.
Reading the shading question
Exam wording is deliberate. "Nearer to A than to B" is one side of the perpendicular bisector. "Within 4 cm of C" is inside a circle. "At least 4 cm" is outside it, and "no more than" is inside including the edge. Draw every boundary first, lightly, then decide which side of each one you want, then shade the region satisfying all of them. Shading before all the boundaries are drawn is how a region ends up one condition short.