M28 · Constructions, loci and bearings | Edexcel GCSE Maths

Revise the key ideas

For a construction, use a ruler for straight lines and a compass for arcs. Leave construction arcs visible; a protractor drawing is not a substitute when ruler-and-compass construction is required.
To bisect a segment AB, use an equal compass radius greater than half AB from A and B. Join the two arc intersections: this perpendicular bisector is at 90° to AB and passes through its midpoint.
Constructing a perpendicular bisector
Enlarge diagram: Constructing a perpendicular bisector
To bisect an angle, draw an arc from the vertex crossing both arms. From those two crossings draw equal-radius arcs that meet inside the angle; join their intersection to the vertex.
To construct a perpendicular from a point on a line, mark equal distances on the line on either side of the point. Construct their perpendicular bisector.
To construct a perpendicular from a point off a line, draw an arc from the point crossing the line twice, then bisect the segment between those crossings. The result passes through the original point.
The shortest distance from a point to a line is measured perpendicular to the line. A sloping connection is longer.

A locus is the set of points meeting a condition. Points a fixed distance from one point form a circle; points equidistant from A and B lie on the perpendicular bisector of AB.
Points equidistant from two intersecting lines lie on their angle bisectors. Points a fixed distance from a straight line lie on two parallel lines, one on each side.
For “within 3 cm of A”, include the circular region of radius 3 cm, not just its circumference. For “more than 3 cm away”, use the outside region with the boundary excluded.
Combine conditions by taking their intersection. A point within 4 cm of A and closer to B than C must lie in the overlap of the circle region and the appropriate half-plane.
Use solid boundaries for “at most” or “at least” when equality is allowed; make excluded boundaries clear if the question uses strict inequalities.

A bearing is measured clockwise from north and written with three digits: east is 090°, south 180° and west 270°. Draw a north line at the starting point.
A bearing of one hundred twenty degrees
Enlarge diagram: A bearing of one hundred twenty degrees
For a reverse bearing, add or subtract 180° and keep the result from 000° to 359°. A bearing of B from A of 065° gives A from B as 245°.
Worked example: At scale 1 cm : 2 km, a 7 km journey on bearing 120° is drawn as a 3.5 cm line clockwise 120° from north. The bearing's starting point matters.
When several bearings are given, use parallel north lines and angle rules to find triangle angles. Do not treat the bearing as automatically an interior angle of the triangle.

Practice questions with explained answers will be added here.

For now, cover the worked answers, try the calculations yourself, then compare each step. Include units and reasons where needed.