GCSE Maths · Loci, Bearings and Vectors · Foundation · Summary

GCSE Maths Loci, Bearings and Vectors (Foundation): summary and key words

Part of the GCSE Maths loci, bearings and vectors chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page

Chapter summary

Skill	Method
Scale: map to real life	Multiply by the scale number (1 : n means 1 cm is n cm); change cm to m (÷ 100) or km (÷ 100 000)
Scale: real life to map	Change to cm, then divide by the scale number
Write a scale as 1 : n	Change both lengths to cm, put the map length first, divide so the first number is 1
Perpendicular bisector	Equal arcs from each end (more than half), join the crossing points
Angle bisector	Arc from the corner, equal arcs from the two crossing points, join the corner to where they cross
Perpendicular from / at a point	Arc from the point to cut the line twice, then equal arcs from those two points
Construct a triangle	SSS: base then two arcs. SAS: side, angle, side. ASA: side then angle at each end
Loci	Circle (fixed distance from a point), perpendicular bisector (equidistant from two points), parallel lines/stadium (distance from a line), angle bisector (two lines)
Combined loci	Draw each locus, then shade only where all the rules overlap; check with a test point
Bearings	Measure from north, clockwise, three figures; draw a north line at every point
Back bearing	Add 180° if the bearing is less than 180°, subtract 180° if it is 180° or more
Two-leg journey	Sketch with north lines, find the angle with parallel-line facts, use Pythagoras/trig (cosine and sine rules in Higher), then change to a bearing
Column vectors	Top = across (right +), bottom = up. Add/subtract each row. Scalar: multiply both numbers. Magnitude: √(x² + y²)
Vector between points	End minus start for each coordinate
Vector geometry (Higher)	Route method: go along known vectors, against an arrow gives a negative; midpoint is half; ratio m/(m + n); parallel = scalar multiple; collinear = parallel + shared point

Can you…?

I can change between map distances and real distances for a scale such as 1 : 50 000.
I can write a scale as a ratio 1 : n from a map distance and a real distance.
I can construct a perpendicular bisector, an angle bisector, a perpendicular from or at a point, and a 60° angle.
I can construct a triangle from SSS, SAS or ASA information.
I can describe the four basic loci and draw them accurately.
I can shade the region that satisfies two or three rules at once.
I can read, draw and write three-figure bearings.
I can find a back bearing and use parallel lines to find angles between bearings.
I can solve two-leg bearings problems with Pythagoras and trigonometry.
I can write a translation as a column vector and use column vectors to add, subtract and scale.
I can find the magnitude of a vector and decide if two vectors are parallel.
Higher: I can use the route method to write vectors in terms of a and b, find midpoints and ratio points, and prove that lines are parallel or that points are collinear.

Key terms

Scale: The ratio of a length on a map or drawing to the real length, e.g. 1 : 50 000.
Construct: Draw accurately using only a ruler and compasses.
Perpendicular bisector: The line that cuts a line segment in half at 90°.
Bisect: To cut into two equal parts.
Angle bisector: The line that cuts an angle into two equal angles.
Locus: The set of all points that obey a rule (plural: loci).
Equidistant: The same distance away.
Bearing: An angle measured clockwise from north, written with three figures.
Back bearing: The bearing of the return direction: the original bearing plus or minus 180°.
Co-interior angles: Angles between two parallel lines on the same side of a crossing line. They add up to 180°.
Vector: A quantity with both size (magnitude) and direction.
Column vector: A vector written as two numbers: across on top, up underneath.
Scalar: An ordinary number, with size but no direction.
Magnitude: The length of a vector, written |a|.
Translation: A move of every point of a shape by the same vector.
Parallel vectors: Vectors that are scalar multiples of each other.
Position vector: A vector giving the position of a point from the origin.
Collinear: Lying on the same straight line.