Geometry and proof · Ages 14–15
Telling the centres apart worksheet
Part of: Centres of a triangle
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Telling the centres apart worksheet
Geometry and proof · Ages 14–15
Numsprite
numsprite.com
Name:Date:Score:／48
Write each answer in the space. For word problems, show your working and write a sentence answer.
Part A: Working out and filling in (2 marks each)
1.In the right triangle ABC, ∠C = 90° and the hypotenuse AB = 7 cm. How far apart are its circumcentre and orthocentre?
Answer:
2.In the right triangle ABC, ∠C = 90° and the hypotenuse AB = 23 cm. How far apart are its circumcentre and orthocentre?
Answer:
Part B: Choose, or true or false (2 marks each)
3.In △ABC, ∠B = 70° and ∠C = 50°. In which picture is P the centroid of △ABC?
- A
- B
- C
- D
Choose the right picture
Answer:
4.In △ABC, ∠B = 75° and ∠C = 60°. In which picture is P the incentre of △ABC?
- A
- B
- C
- D
Choose the right picture
Answer:
5.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:
6.In △ABC, ∠B = 65° and ∠C = 55°. In which picture is P the incentre of △ABC?
- A
- B
- C
- D
Choose the right picture
Answer:
7.Which two centres are always inside the triangle?
- A. the incentre and the centroid
- B. the centroid and the orthocentre
- C. the circumcentre and the orthocentre
- D. the incentre and the circumcentre
Choose the right answer
Answer:
8.Which centre of △ABC is the point P in the picture?
- A. circumcentre
- B. incentre
- C. orthocentre
- D. centroid
Choose the right answer
Answer:
9.For an equilateral triangle with 8 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. they are four different points
- D. only the incentre and circumcentre coincide
Choose the right answer
Answer:
10.For an equilateral triangle with 10 cm sides, how are its incentre, circumcentre, centroid and orthocentre related?
- A. the circumcentre is outside
- B. they are four different points
- C. only the incentre and circumcentre coincide
- D. all four are one and the same point
Choose the right answer
Answer:
11.△ABC's angles are 100°, 30° and 50°. Which of its four centres lie outside it?
- A. none: all four are inside
- B. the circumcentre and the orthocentre
- C. the incentre and the centroid
- D. only the circumcentre
Choose the right answer
Answer:
12.△ABC's angles are 115°, 20° and 45°. Which of its four centres lie outside it?
- A. only the circumcentre
- B. none: all four are inside
- C. the circumcentre and the orthocentre
- D. the incentre and the centroid
Choose the right answer
Answer:
Part C: Word problems (4 marks each)
13.An equilateral triangular tile has medians of 15 cm. Its incentre, circumcentre, centroid and orthocentre are one point P. How far is P from each corner?
Working:
Answer:
14.A sprinkler on a triangular lawn is to be 4 m from each of its three edges. At which centre should it go?
- A. incentre
- B. centroid
- C. circumcentre
- D. orthocentre
Choose the right answer
Working:
Answer:
15.A hospital is to be built 8 km from each of three towns. At which centre of the towns' triangle should it go?
- A. orthocentre
- B. incentre
- C. centroid
- D. circumcentre
Choose the right answer
Working:
Answer:
16.A triangular card about 36 cm across is to balance on a pencil point. Under which centre should the point go?
- A. centroid
- B. orthocentre
- C. incentre
- D. circumcentre
Choose the right answer
Working:
Answer:
17.A triangular card about 21 cm across is to balance on a pencil point. Under which centre should the point go?
- A. centroid
- B. circumcentre
- C. orthocentre
- D. incentre
Choose the right answer
Working:
Answer:
18.An equilateral triangular tile has medians of 18 cm. Its incentre, circumcentre, centroid and orthocentre are one point P. How far is P from each corner?
Working:
Answer:
Show the answers and working
Telling the centres apart worksheet: Answers and working
Part A: Working out and filling in
1. 3.5 cm
The circumcentre O is the middle of AB and the orthocentre is the right-angle vertex C; OC = OA = 7 ÷ 2 = 3.5 cm (the circumcentre is equally far from the vertices).
2. 11.5 cm
The circumcentre O is the middle of AB and the orthocentre is the right-angle vertex C; OC = OA = 23 ÷ 2 = 11.5 cm (the circumcentre is equally far from the vertices).
Part B: Choose, or true or false
3. C
The centroid comes from the medians: lines from the vertices to the opposite sides' middles, each side's halves ticked, no right angles.
4. B
The incentre comes from the three angle bisectors: each angle cut into equal halves with matching arcs.
5. A
The centroid comes from the medians: lines from the vertices to the opposite sides' middles, each side's halves ticked, no right angles.
6. D
The incentre comes from the three angle bisectors: each angle cut into equal halves with matching arcs.
7. A. 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.
8. C. 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.
9. B. all four are one and the same point
An equilateral triangle is symmetrical: each bisector is also a median, a height and a perpendicular bisector, so the four centres are one point.
10. D. all four are one and the same point
An equilateral triangle is symmetrical: each bisector is also a median, a height and a perpendicular bisector, so the four centres are one point.
11. B. the circumcentre and the orthocentre
100° is obtuse: the circumcentre and orthocentre lie outside, while the incentre and centroid stay inside.
12. C. the circumcentre and the orthocentre
115° is obtuse: the circumcentre and orthocentre lie outside, while the incentre and centroid stay inside.
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. 10 cm
P is the centroid, 23 of a median from a corner: 15 × 23 = 10 cm (as the circumcentre, it is the same distance from all three).
14. A. incentre
The point equally far from three sides is the incentre (where the angle bisectors meet).
15. D. circumcentre
The point equally far from three corners is the circumcentre (where the perpendicular bisectors meet).
16. A. centroid
The centroid (where the medians meet) is the balance point: each median halves the area.
17. A. centroid
The centroid (where the medians meet) is the balance point: each median halves the area.
18. 12 cm
P is the centroid, 23 of a median from a corner: 18 × 23 = 12 cm (as the circumcentre, it is the same distance from all three).
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