MathematicsChapter 10: Practical Geometry
Angle Bisector
An angle bisector is the ray that splits an angle into two equal angles.
Learning objectives
- Explain angle bisector in clear mathematical language.
- Use the main rule, formula, test, or construction correctly.
- Work through a simple example and show each important step.
- Identify common mistakes and write an exam-ready conclusion.
Definition
An angle bisector is the ray that splits an angle into two equal angles. To draw it: open a compass at the vertex and cut both arms with the same arc; from those two cut points draw equal arcs that meet; the line from the vertex through that meeting point is the bisector. In a triangle the three angle bisectors are concurrent. Their common point is the in-centre. The in-centre is always inside the triangle and is equidistant from the sides, so it is the centre of the incircle (the circle that touches all three sides).
Where / symbols
- equal angles
- An angle bisector is the ray that splits an angle into two equal angles.
- cut both arms
- To draw it: open a compass at the vertex and cut both arms with the same arc; from those two cut points draw equal arcs that meet; the line from the vertex through that meeting point is the bisector.
- equal arcs
- To draw it: open a compass at the vertex and cut both arms with the same arc; from those two cut points draw equal arcs that meet; the line from the vertex through that meeting point is the bisector.
- concurrent
- In a triangle the three angle bisectors are concurrent.
- in-centre
- Their common point is the in-centre.
- always inside
- The in-centre is always inside the triangle and is equidistant from the sides, so it is the centre of the incircle (the circle that touches all three sides).
Overview
Construction on one side of the page, the meeting point on the other:
Drawing the Bisector
Use the same radius from the vertex to cut arm XP and arm XQ at P and Q. Then, still using equal radii (they need not equal the first radius), swing an arc from P and from Q. Let the arcs meet at R. Ray XR bisects angle PXQ. The equal-arc step is what forces the two smaller angles to match.
Examples Bisect a 70 deg angle into two 35 deg angles · Do not change the first compass opening until both arms are cut · The meeting of the second pair of arcs is not on the arms
In-Centre and Incircle
Drop a perpendicular from the in-centre to any side: that length is the in-radius. The same length reaches the other two sides, which is why one circle can touch all three. Because each bisector stays inside its angle, the three bisectors always meet inside, even in an obtuse triangle. That is different from the orthocentre and the circumcentre, which can go outside.
Examples In-centre of an equilateral triangle is also the other centres · In a scalene triangle the in-centre is still inside · The incircle touches each side at one point, not at a vertex
WHAT — Core idea
An angle bisector is the ray that splits an angle into two equal angles. To draw it: open a compass at the vertex and cut both arms with the same arc; from those two cut points draw equal arcs that meet; the line from the vertex through that meeting point is the bisector. In a triangle the three angle bisectors are concurrent. Their common point is the in-centre. The in-centre is always inside the triangle and is equidistant from the sides, so it is the centre of the incircle (the circle that touches all three sides).
HOW — Rules, method, and examples
1. DRAWING THE BISECTOR: Use the same radius from the vertex to cut arm XP and arm XQ at P and Q. Then, still using equal radii (they need not equal the first radius), swing an arc from P and from Q. Let the arcs meet at R. Ray XR bisects angle PXQ. The equal-arc step is what forces the two smaller angles to match. 2. IN-CENTRE AND INCIRCLE: Drop a perpendicular from the in-centre to any side: that length is the in-radius. The same length reaches the other two sides, which is why one circle can touch all three. Because each bisector stays inside its angle, the three bisectors always meet inside, even in an obtuse triangle. That is different from the orthocentre and the circumcentre, which can go outside.
WHY — Meaning and connections
Equidistant from the sides means perpendicular distance, not distance to the vertices. Distance to the vertices is the circumcentre's job. Mixing in-centre with circumcentre is a frequent exam slip.
HOW — Answer in an exam
Bisector construction: one arc cuts both arms, then equal arcs from those points. Three bisectors meet at the in-centre, always inside, equidistant from the sides. That point is the centre of the incircle.
Explanation
Equidistant from the sides means perpendicular distance, not distance to the vertices. Distance to the vertices is the circumcentre's job. Mixing in-centre with circumcentre is a frequent exam slip.
In simple words
Bisector construction: one arc cuts both arms, then equal arcs from those points. Three bisectors meet at the in-centre, always inside, equidistant from the sides. That point is the centre of the incircle.
Key points
- equal angles
- cut both arms
- equal arcs
- concurrent
- same radius
- equal radii
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Angle Bisector
Sticky Notes · Quick Revision
WHAT
- An angle bisector is the ray that splits an angle into two equal angles.
Key Points
- An angle bisector is the ray that splits an angle into two equal angles.
- Use the same radius from the vertex to cut arm XP and arm XQ at P and Q.
- Drop a perpendicular from the in-centre to any side: that length is the in-radius.
- Equidistant from the sides means perpendicular distance, not distance to the vertices.
HOW
- 1.
- Use the same radius from the vertex to cut arm XP and arm XQ at P and Q.
- Drop a perpendicular from the in-centre to any side: that length is the in-radius.
Examples
- Bisect a 70 deg angle into two 35 deg angles
- Do not change the first compass opening until both arms are cut
- The meeting of the second pair of arcs is not on the arms
- In-centre of an equilateral triangle is also the other centres
- In a scalene triangle the in-centre is still inside
Common Mistakes
- Correction: Bisector construction: one arc cuts both arms, then equal arcs from those points.
- Correction: Three bisectors meet at the in-centre, always inside, equidistant from the sides.
- Correction: That point is the centre of the incircle.
Exam Tip
- Bisector construction: one arc cuts both arms, then equal arcs from those points. Three bisectors meet at the in-centre, always inside, equidistant from the sides. That point is the centre of the incircle.
60-second revision
- An angle bisector is the ray that splits an angle into two equal angles.
- Use the same radius from the vertex to cut arm XP and arm XQ at P and Q.
- Drop a perpendicular from the in-centre to any side: that length is the in-radius.
- Equidistant from the sides means perpendicular distance, not distance to the vertices.
- Bisector construction: one arc cuts both arms, then equal arcs from those points.
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