Geometry and proof · Ages 14–15

The circumcentre worksheet

Part of: Centres of a triangle

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Geometry and proof · Ages 14–15

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Write each answer in the space. For word problems, show your working and write a sentence answer.

1.In △ABC, ∠C = 90° and the hypotenuse AB = 23 cm; O is the circumcentre. OC =
Answer:
2.O is the circumcentre of △ABC, and OA = 13 cm. OC =
Answer:
3.In △ABC, ∠C = 90° and the hypotenuse AB = 18 cm; O is the circumcentre. OC =
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4.O is the circumcentre of △ABC, and OA = 6 cm. OC =
Answer:
5.O is the circumcentre of the acute triangle ABC, with ∠OAB = 43° and ∠OBC = 30°. ∠OBA = °, ∠OCB = °
Fill in each blank
Answer:
6.In the picture, O is the circumcentre of △ABC. x =
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7.O is the circumcentre of the acute triangle ABC, with ∠OAB = 47° and ∠OBC = 41°. ∠OBA = °, ∠OCB = °
Fill in each blank
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8.In △ABC, ∠C = 90°, AC = 21 cm and BC = 72 cm. The circumcircle's radius =
Answer:
9.In △ABC, ∠C = 90°, AC = 3 cm and BC = 4 cm. The circumcircle's radius =
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13.Three houses A, B and C form a right triangle with ∠C = 90°, AC = 88 m and BC = 66 m. A mast is to be equally far from all three. AB = m, and the mast is m from each house.
Fill in each blank
Working:
Answer:
14.Three trees A, B and C form an acute triangle, and a swing O is equally far from all three. ∠OAB = 21° and ∠OBC = 19°. ∠OCA = °, ∠ABC = °
Fill in each blank
Working:
Answer:
15.A triangular flower stand's circumcentre O is 38 cm from its corner A. A round frame is made through its three corners. What is the frame's diameter?
Working:
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16.A right-triangular wooden frame has a hypotenuse of 34 cm. A round card is cut to pass exactly through its three corners. What is the circle's radius?
Working:
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17.A triangular flower stand's circumcentre O is 10 cm from its corner A. A round frame is made through its three corners. What is the frame's diameter?
Working:
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18.A right-triangular wooden frame has a hypotenuse of 91 cm. A round card is cut to pass exactly through its three corners. What is the circle's radius?
Working:
Answer:

Show the answers and working

The circumcentre worksheet: Answers and working

1. 11.5 cm
A right triangle's circumcentre is the middle of its hypotenuse: OC = OA = OB = 23 ÷ 2 = 11.5 cm.
2. 13 cm
The circumcentre is equally far from all three vertices (⊥ bisector theorem): OC = OA = 13 cm.
3. 9 cm
A right triangle's circumcentre is the middle of its hypotenuse: OC = OA = OB = 18 ÷ 2 = 9 cm.
4. 6 cm
The circumcentre is equally far from all three vertices (⊥ bisector theorem): OC = OA = 6 cm.
5. 43, 30
OA = OB = OC, so △OAB and △OBC are isosceles: ∠OBA = ∠OAB = 43° and ∠OCB = ∠OBC = 30°.
6. 25°
OA = OB (the circumcentre is equally far from the vertices), so △OAB is isosceles: x = ∠OAB = 25° (base ∠s, isos. △).
7. 47, 41
OA = OB = OC, so △OAB and △OBC are isosceles: ∠OBA = ∠OAB = 47° and ∠OCB = ∠OBC = 41°.
8. 37.5 cm
AB = 21² + 72² = 75 cm (Pyth. thm.); the circumcentre is the hypotenuse's middle, so the radius = 75 ÷ 2 = 37.5 cm.
9. 2.5 cm
AB = 3² + 4² = 5 cm (Pyth. thm.); the circumcentre is the hypotenuse's middle, so the radius = 5 ÷ 2 = 2.5 cm.

10. D. the three sides' perpendicular bisectors
The circumcentre is where the sides' perpendicular bisectors meet, equally far from the three vertices.
11. A. inside the triangle
Look at the largest angle: acute, the circumcentre is inside; right, at the middle of the hypotenuse; obtuse, outside.
12. B. outside the triangle
Look at the largest angle: acute, the circumcentre is inside; right, at the middle of the hypotenuse; obtuse, outside.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. 110, 55
AB = 88² + 66² = 110 m; the point equally far from three corners is the circumcentre, at a right triangle's hypotenuse's middle: 110 ÷ 2 = 55 m.
14. 50, 40
OA = OB = OC: three isosceles triangles with base angles 21°, 19° and ∠OCA, which make the triangle's angles in pairs: 2(21 + 19 + ∠OCA) = 180, ∠OCA = 50°; ∠ABC = 21° + 19° = 40°.
15. 76 cm
OA = OB = OC = 38 cm is the circumcircle's radius; diameter = 2 × 38 = 76 cm.
16. 17 cm
The circle through the three corners is centred at the circumcentre, the hypotenuse's middle for a right triangle: radius = 34 ÷ 2 = 17 cm.
17. 20 cm
OA = OB = OC = 10 cm is the circumcircle's radius; diameter = 2 × 10 = 20 cm.
18. 45.5 cm
The circle through the three corners is centred at the circumcentre, the hypotenuse's middle for a right triangle: radius = 91 ÷ 2 = 45.5 cm.

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