Angle Bisectors and Incircle
An angle bisector cuts an angle into two equal halves. In a triangle, the three bisectors meet at the incentre. That point is the centre of the incircle: the circle inside the triangle that touches all three sides.
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What Is an Angle Bisector?
- An angle bisector splits an angle into two equal parts.
- Each new angle is exactly half of the original angle.
How to Draw an Angle Bisector?
- You can draw an angle bisector using a protractor by halving the angle.
- You can also use a compass to draw the bisector without measuring.
The Key Property of an Angle Bisector
- Any point on an angle bisector is equidistant from both sides of the angle.
- This means the perpendicular distance to each side is the same.
Incentre and Incircle
- The incentre is where the three angle bisectors of a triangle meet.
- The incircle is the largest circle that fits inside the triangle and touches all three sides.
Practice Questions
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What is an angle bisector?
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An angle bisector is a line or ray that divides an angle into two equal parts.
What tool is commonly used to construct an angle bisector?
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To construct an angle bisector, you can use a protractor to measure the angle, or a compass to draw arcs and bisect the angle accurately.
What is the incentre of a triangle?
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The incentre is the point where the angle bisectors of a triangle intersect.
What is the inradius of a triangle?
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The inradius is the distance from the incentre to any side of the triangle. This distance is the same for all three sides.
If you want to draw the incircle of a triangle, what do you need to know?
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To draw the incircle, you need to know the location of the incentre (where the angle bisectors meet) and the inradius (the distance from the incentre to any side).
To find the largest circle that you can cut out of a triangle piece of paper, what should you do?
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To find the largest circle that fits inside a triangle, you need to find the incircle. This is done by drawing the angle bisectors to locate the incentre, then measuring the distance from it to any side to determine the inradius.
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Students Also Ask
The questions students bump into most on this topic
For a 40° angle, the angle bisector passes through the 20° mark, splitting the original angle into two equal 20° parts. You draw a ray from the vertex at exactly half the measured angle. This works for any angle, whether acute, obtuse, or reflex.
Every point on an angle bisector sits the same distance from both sides of the angle. This equidistant property means that if you measure the perpendicular distance from any point on the bisector to each side, the two distances are always equal. This property is used to locate the incentre of a triangle.
No. If drawn accurately, the intersection of just two angle bisectors gives you the incentre. The third bisector serves as a useful check. If all three bisectors do not meet at a single point, at least one of your constructions contains an error that needs correcting.
The inradius is the perpendicular distance from the incentre to any side of the triangle. Because the incentre is equidistant from all three sides, this distance is the same no matter which side you measure to. The inradius is also the radius of the incircle.
Yes. Every triangle has exactly one incentre and therefore exactly one incircle. The incircle touches all three sides without crossing them and is the largest circle that fits entirely inside the triangle, regardless of whether the triangle is scalene, isosceles, or equilateral.