The ruler-and-compass constructions, what each one gives, and when a triangle can be drawn at all.
What you will learn
- Know the steps of the standard constructions
- Know which angles can be constructed
- Decide if three lengths make a triangle
The constructions
To bisect means to cut into two equal halves.
Perpendicular means at right angles (90°). A construction uses only a ruler (straight edge) and compasses: no protractor, and no measuring the middle by eye.
Perpendicular bisector of AB: the line at right angles to AB through its midpoint.
- with centre A, draw arcs above and below AB, with radius more than half of AB
- with centre B and the same radius, draw arcs that cross the first two
- join the two crossing points with a straight line
Every point on it is the same distance from A and B, because each crossing point is one radius from A and one radius from B.
Angle bisector of angle ABC (the angle at B):
- with centre B, draw an arc crossing BA and BC
- from each of those two crossing points, draw equal arcs that cross each other
- join that crossing point to B
The bisector splits the angle into two equal angles, so bisecting 70° gives two angles of 35°. Every point on it is the same distance from the two arms BA and BC.
A 60° angle: with centre A and then centre B, draw arcs of radius AB that cross at C. Then CA = CB = AB, so triangle ABC is equilateral and each of its angles is 60°.
Perpendicular from a point P to a line: the shortest path from P to the line.
From 60° (equilateral triangle) and 90° (perpendicular), bisecting gives 30°, 45° and 15°, so those can be constructed too. An angle such as 50° cannot: it needs a protractor.
Leave the construction arcs showing: on a paper they earn the marks.
A locus (plural loci) is the set of all points that follow a rule. Points the same distance from A and B lie on the perpendicular bisector of AB. Points the same distance from two lines AB and AC lie on the bisector of angle BAC. Points a fixed distance from one point make a circle. Points a fixed distance, say 2 cm, from a straight line make two lines parallel to it, 2 cm either side.
The perpendicular bisector crosses AB at its midpoint. With coordinates, the midpoint is the mean of the x values and the mean of the y values: A(2, 5) and B(6, −1) have midpoint ((2 + 6) ÷ 2, (5 + (−1)) ÷ 2) = (4, 2).
The rest of this lesson and its 7 practice questions are locked.
In this lesson
- The constructions
- Can it be a triangle?
- Try it: Construct and check
7 practice questions, marked as you go; about 20 minutes in all.
Key terms
Bisect, Perpendicular, Perpendicular bisector, Angle bisector, Perpendicular from a point, Locus, Loci, Midpoint
In this unit: Constructions, loci and bearings
Three-figure bearings, scale drawings and maps, ruler-and-compass constructions, and loci: sets of points that follow a rule.
- Bearings
- Scale drawings and maps
- Constructions
- Loci