Geometrical constructions
Section: Geometry | Syllabus: Cambridge IGCSE Mathematics (0580)
Constructing a Triangle from Its Side Lengths
A triangle can be constructed accurately using only a ruler and a pair of compasses, when the lengths of all three sides are known.
- Draw the longest side first as a straight line, measured accurately with a ruler
- Open the compasses to the length of a second side, and draw an arc from one end of the line
- Open the compasses to the length of the third side, and draw an arc from the other end of the line - the point where the two arcs cross is the third vertex
- Join both ends of the original line to this new point to complete the triangle
Worked Example: Constructing a Triangle from Three Side Lengths
- Question: Construct a triangle with sides 5 cm, 7 cm and 9 cm.
- Step 1: Draw the longest side, 9 cm, as a straight line using a ruler
- Step 2: Open the compasses to 5 cm and draw an arc from one end of the 9 cm line
- Step 3: Open the compasses to 7 cm and draw an arc from the other end of the 9 cm line
- Step 4: Join the intersection of the two arcs to both ends of the 9 cm line
- Answer: triangle constructed with sides 5 cm, 7 cm and 9 cm
Common Mistakes
MistakeErasing the construction arcs after finishing, leaving no evidence of the method used
Fixconstruction arcs must be left visible on the diagram - they are part of what is being assessed, not just scrap working
MistakeMeasuring a side length inaccurately with the ruler, which throws off the whole triangle
Fixmeasure the base line as precisely as possible before opening the compasses, since every other measurement depends on it
Constructing a Perpendicular Bisector
A perpendicular bisector is a line that cuts a given line segment exactly in half, at a right angle. It can be constructed accurately using only compasses and a straight edge.
- Open the compasses to more than half the length of the line segment
- With the point on one end of the line, draw an arc above and below the line; repeat from the other end, without changing the compass width
- Draw a straight line through the two points where the arcs cross - this is the perpendicular bisector
- Every point on the perpendicular bisector is exactly the same distance from both ends of the original line segment
Worked Example: Constructing a Perpendicular Bisector
- Question: Construct the perpendicular bisector of a line segment AB, 8 cm long.
- Step 1: Open the compasses to a width of more than 4 cm (more than half of AB)
- Step 2: With the compass point on A, draw arcs above and below the line; repeat from B with the same width
- Step 3: Draw a straight line through the two points where the arcs intersect
- Answer: perpendicular bisector constructed, crossing AB at its midpoint (4 cm from each end) at a right angle
Common Mistakes
MistakeOpening the compasses to exactly half the length of the line (or less), so the arcs from each end don't cross at all
Fixthe compass width must be more than half the line's length, so the arcs drawn from each end overlap
MistakeChanging the compass width between drawing arcs from A and from B
Fixkeep the compasses at the same fixed width throughout - only the centre point (A, then B) should move
Constructing an Angle Bisector
An angle bisector is a line that splits a given angle exactly in half. It can be constructed accurately using only compasses and a straight edge.
- With the compass point on the vertex of the angle, draw an arc that crosses both lines forming the angle
- From each point where this arc crosses a line, draw a second arc into the middle of the angle, using the same compass width for both
- Draw a straight line from the vertex through the point where these two new arcs cross - this is the angle bisector
- Every point on the angle bisector is exactly the same distance from both lines forming the angle
Worked Example: Constructing an Angle Bisector
- Question: Construct the bisector of angle ABC.
- Step 1: With the compass point on vertex B, draw an arc crossing both arms BA and BC
- Step 2: From each of these two crossing points, draw a second arc into the middle of the angle, keeping the same compass width
- Step 3: Draw a straight line from B through the point where the two new arcs intersect
- Answer: angle bisector constructed, splitting angle ABC into two equal angles
Common Mistakes
MistakeChanging the compass width when drawing the second pair of arcs
Fixthe same fixed compass width should be used for the second pair of arcs, so that the resulting point is genuinely equidistant from both lines
MistakeConfusing an angle bisector with a perpendicular bisector, and constructing the wrong one
Fixan angle bisector starts from the vertex and splits an angle; a perpendicular bisector cuts across a line segment at its midpoint - they use similar arc techniques for different purposes
Loci Involving Points and Lines
A locus is the set of all points that satisfy a given rule, such as being a fixed distance from a point or a line. Constructing a locus draws this whole set of points at once.
- The locus of points a fixed distance from a given point is a circle, centred on that point, with the given distance as its radius
- The locus of points a fixed distance from a given straight line segment is a pair of parallel lines, one on each side, joined by semicircles at each end (a "racetrack" shape)
- The locus of points equidistant from two given points is the perpendicular bisector of the line segment joining them
- The locus of points equidistant from two given intersecting straight lines is the pair of lines that bisect the angles between them
Worked Example: Describing a Locus
- Question: A goat is tied to a post by a rope 3 m long, in an open field. Describe the locus of points the goat can reach.
- Step 1: The goat can move in any direction, always staying within 3 m of the post
- Step 2: This matches the locus of points a fixed distance from a single point
- Answer: a circle of radius 3 m, centred on the post
Common Mistakes
MistakeDrawing the locus of points equidistant from two points as a circle, confusing it with the "fixed distance from one point" locus
Fixequidistant from two points always means the perpendicular bisector of the segment joining them - a straight line, not a circle
MistakeForgetting the rounded (semicircular) ends when constructing the locus of points a fixed distance from a line segment
Fixfor a locus around a finite line segment, add a semicircular arc of the correct radius at each end, not just two straight parallel lines
Interactive revision notes, videos and practice questions load below.