Centres of a triangle · Ages 14–15
Telling the centres apart
Bisectors → incentre; ⊥ bisectors → circumcentre; medians → centroid; heights → orthocentre.
How to do it
Method
Bisectors → incentre; ⊥ bisectors → circumcentre; medians → centroid; heights → orthocentre
Right triangle: circumcentre at the hypotenuse's middle, orthocentre at the right angle
Another way
Read the marks
Right-angle marks at the sides' middles: perpendicular bisectors, circumcentre
In full
Each centre comes from its own lines: the incentre from the three angle bisectors (matching arcs in the angles); the circumcentre from the three perpendicular bisectors (right-angle marks at the sides' middles); the centroid from the three medians (each side cut into two equal parts, the line from the opposite vertex); the orthocentre from the three heights (from a vertex, at right angles to the opposite side). The method: see whether the lines start at the vertices or at the sides' middles, then look for right-angle marks and equal-angle marks. Special triangles: an equilateral triangle's four centres are one point; an isosceles triangle's all lie on its line of symmetry; a right triangle's circumcentre is the middle of its hypotenuse and its orthocentre the right-angle vertex; an obtuse triangle's circumcentre and orthocentre lie outside. The incentre and the centroid are always inside.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
Which centre of △ABC is the point P in the picture?
- A. orthocentre
- B. incentre
- C. circumcentre
- D. centroid
Choose the right answer
Show the answer and working
Answer: A. orthocentre
See where the lines start and how they're marked: matching arcs mean angle bisectors, right angles at the sides' middles perpendicular bisectors, ticked halves medians, right angles at the far side from a vertex heights.
Example 2
Which two centres are always inside the triangle?
- A. the centroid and the orthocentre
- B. the incentre and the centroid
- C. the incentre and the circumcentre
- D. the circumcentre and the orthocentre
Choose the right answer
Show the answer and working
Answer: B. the incentre and the centroid
Angle bisectors and medians run from the vertices into the triangle, so they meet inside; the circumcentre and orthocentre can be outside.
Example 3
In the right triangle ABC, ∠C = 90° and the hypotenuse AB = 6 cm. How far apart are its circumcentre and orthocentre?
Show the answer and working
Answer: 3 cm
The circumcentre O is the middle of AB and the orthocentre is the right-angle vertex C; OC = OA = 6 ÷ 2 = 3 cm (the circumcentre is equally far from the vertices).
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.In △ABC, ∠B = 60° and ∠C = 45°. In which picture is P the incentre of △ABC?
- A
- B
- C
- D
Choose the right picture
Answer:
2.In △ABC, ∠B = 55° and ∠C = 60°. In which picture is P the centroid of △ABC?
- A
- B
- C
- D
Choose the right picture
Answer:
3.For an equilateral triangle with 3 cm sides, how are its incentre, circumcentre, centroid and orthocentre related?
- A. the circumcentre is outside
- B. all four are one and the same point
- C. only the incentre and circumcentre coincide
- D. they are four different points
Choose the right answer
Answer:
4.△ABC's angles are 125°, 25° and 30°. Which of its four centres lie outside it?
- A. the circumcentre and the orthocentre
- B. only the circumcentre
- C. none: all four are inside
- D. the incentre and the centroid
Choose the right answer
Answer:
5.In the right triangle ABC, ∠C = 90° and the hypotenuse AB = 23 cm. How far apart are its circumcentre and orthocentre?
Answer:
All the answers
- 1. D
- 2. C
- 3. B. all four are one and the same point
- 4. A. the circumcentre and the orthocentre
- 5. 11.5 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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