IGCSE Maths / Geometric constructions
Constructing the bisector of an angle
Edexcel International GCSE Maths (4MA1), chapter 13: Geometric constructions, specification point 4.5d. Free lesson.
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An angle bisector is a line from the corner of an angle that divides it into two equal angles. To construct one with a ruler and compasses, use three visible arcs and join the corner to where the last two arcs cross.
What an angle bisector does
To bisect means to split something into two equal parts. An angle bisector starts at the corner of the angle and divides it into two equal angles. Every point on the bisector is the same distance from the two arms. The arcs are part of the construction and must stay visible. The finished line must start at the corner and pass through the crossing of the last two arcs.
How to construct the bisector
Use a ruler and compasses only. The first arc marks matching points on the two arms. Then a matching pair of arcs from those points gives you the point to join to the corner.
Why the line splits the angle evenly
In the construction, OP = OQ because both lengths come from the first arc. Also, PR = QR because the two arcs from P and Q use the same compass width. These matching lengths make the shape symmetrical about OR, so the angles on either side of OR are equal. For an angle AOB of 70°, the two equal halves are each 70° ÷ 2 = 35°. The arcs from P and Q must have the same width. The first arc can have a different width, as long as it reaches both arms.
Choose the correct line and find an angle
There are three marked points, P, Q and R, so check carefully which points you join. PQ joins the points where the first arc cut the arms. It crosses the angle, but it does not bisect it. The bisector is OR, from the corner through R. If OR bisects an 84° angle, each half is 84° ÷ 2 = 42°.