O is the circumcentre of the acute triangle ABC, with ∠OAB = 26° and ∠OBC = 48°. ∠OBA = □°, ∠OCB = □° Fill in each blank

26, 48. OA = OB = OC, so △OAB and △OBC are isosceles: ∠OBA = ∠OAB = 26° and ∠OCB = ∠OBC = 48°.

△ABC's angles are 40°, 70° and 70°. Where is its circumcentre? A. at one of the vertices B. at the middle of the longest side C. outside the triangle D. inside the triangle Choose the right answer

D. inside the triangle. Look at the largest angle: acute, the circumcentre is inside; right, at the middle of the hypotenuse; obtuse, outside.

Centres of a triangle · Ages 14–15

The circumcentre

Circumcentre: where the perpendicular bisectors meet, equally far from the three vertices.

How to do it

Method

Circumcentre: where the perpendicular bisectors meet, equally far from the three vertices

OA = OB = OC; right triangle, hypotenuse 10: OA = 10 ÷ 2 = 5

Another way

Isosceles triangles

OA = OB: ∠OAB = ∠OBA

In full

The perpendicular bisectors of a triangle's three sides meet at one point O, the circumcentre. O lies on each, so it is equally far from all three vertices (⊥ bisector theorem): OA = OB = OC, and a circle centred at O passes through all three, the circumcircle. Where it lies: inside an acute triangle; at the middle of the hypotenuse of a right-angled one; outside an obtuse one. Chasing angles: OA = OB makes △OAB isosceles, so ∠OAB = ∠OBA. Another way: a right triangle's hypotenuse is the circumcircle's diameter, so with a hypotenuse of 10 cm, OA = OB = OC = 5 cm.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
In the picture, O is the circumcentre of △ABC. x =
Show the answer and working
Answer: 40°
OA = OB (the circumcentre is equally far from the vertices), so △OAB is isosceles: x = ∠OAB = 40° (base ∠s, isos. △).
Example 2
O is the circumcentre of the acute triangle ABC, with ∠OAB = 26° and ∠OBC = 48°. ∠OBA = °, ∠OCB = °
Fill in each blank
Show the answer and working
Answer: 26, 48
OA = OB = OC, so △OAB and △OBC are isosceles: ∠OBA = ∠OAB = 26° and ∠OCB = ∠OBC = 48°.
Example 3
△ABC's angles are 40°, 70° and 70°. Where is its circumcentre?
- A. at one of the vertices
- B. at the middle of the longest side
- C. outside the triangle
- D. inside the triangle
Choose the right answer
Show the answer and working
Answer: D. inside the triangle
Look at the largest angle: acute, the circumcentre is inside; right, at the middle of the hypotenuse; obtuse, outside.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.In △ABC, ∠C = 90° and the hypotenuse AB = 28 cm; O is the circumcentre. OC =
Answer:
2.O is the circumcentre of the acute triangle ABC, with ∠OAB = 24° and ∠OBC = 46°. ∠OBA = °, ∠OCB = °
Fill in each blank
Answer:
3.Where do which three lines meet to give a triangle's circumcentre?
- A. the three medians
- B. the three heights
- C. the three angle bisectors
- D. the three sides' perpendicular bisectors
Choose the right answer
Answer:
4.△ABC's angles are 50°, 30° and 100°. Where is its circumcentre?
- A. inside the triangle
- B. at one of the vertices
- C. at the middle of the longest side
- D. outside the triangle
Choose the right answer
Answer:
5.In △ABC, ∠C = 90°, AC = 3 cm and BC = 4 cm. The circumcircle's radius =
Answer:

All the answers

1. 14 cm
2. 24, 46
3. D. the three sides' perpendicular bisectors
4. D. outside the triangle
5. 2.5 cm

Free worksheet: The circumcentreWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

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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