Centres of a triangle · Ages 14–15
The incentre
Incentre: where the angle bisectors meet, equally far from the three sides.
How to do it
Method
Incentre: where the angle bisectors meet, equally far from the three sides
∠IBC = 30°, ∠ICB = 40°: ∠BIC = 180° − 70° = 110°
Another way
∠BIC = 90° + ∠A ÷ 2
∠A = 40°: 90° + 20° = 110°
In full
The bisectors of a triangle's three angles meet at one point I, the incentre. I lies on each bisector, so it is equally far from all three sides (∠ bisector theorem); a circle centred at I with that radius touches all three sides, the incircle. The incentre is always inside the triangle. Chasing angles: ∠ABC = 60° and ∠ACB = 80° give ∠IBC = 30° and ∠ICB = 40°, so ∠BIC = 180° − 30° − 40° = 110°. Another way: ∠BIC = 90° + ∠A ÷ 2. ∠A = 180° − 60° − 80° = 40°, and 90° + 20° = 110°, the same. The incircle's radius r: area = r × perimeter ÷ 2.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
I is the incentre of △ABC, 9 cm from BC. Its distance from AB =
Show the answer and working
Answer: 9 cm
The incentre is equally far from all three sides (∠ bisector theorem): 9 cm.
Example 2
In the picture, I is the incentre of △ABC. x =
Show the answer and working
Answer: 125°
In △IBC: x = 180° − 29° − 26° = 125° (∠ sum of △).
Example 3
Where do which three lines meet to give a triangle's incentre?
- A. the three medians
- B. the three angle bisectors
- C. the three sides' perpendicular bisectors
- D. the three heights
Choose the right answer
Show the answer and working
Answer: B. the three angle bisectors
The incentre is where the angle bisectors meet, equally far from the three sides.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.In the picture, I is the incentre of △ABC. x =
Answer:
2.I is the incentre of △ABC, 2 cm from BC. Its distance from AB =
Answer:
3.In the picture, I is the incentre of △ABC. x =
Answer:
4.Where do which three lines meet to give a triangle's incentre?
- A. the three sides' perpendicular bisectors
- B. the three angle bisectors
- C. the three heights
- D. the three medians
Choose the right answer
Answer:
5.In △ABC, ∠C = 90°, AC = 15 cm, BC = 36 cm and AB = 39 cm. The incircle's radius =
Answer:
All the answers
- 1. 28°
- 2. 2 cm
- 3. 127°
- 4. B. the three angle bisectors
- 5. 6 cm
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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