Geometry and proof · Ages 14–15
Centres of a triangle worksheet
Covers: The angle bisector's property, The perpendicular bisector's property, The incentre, The circumcentre, The centroid, The orthocentre, Telling the centres apart
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Centres of a triangle worksheet
Geometry and proof · Ages 14–15
Numsprite
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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 picture, PM is the perpendicular bisector of AB. x =
Answer:
2.P is on the perpendicular bisector of the segment AB, with PA = (2x + 1) cm and PB = (3x − 1) cm. x =
Answer:
3.I is the incentre of △ABC, 4 cm from BC. Its distance from AB =
Answer:
4.I is the incentre of △ABC, with ∠ABC = 52° and ∠ACB = 46°. ∠IBC = °, ∠ICB = °, ∠BIC = °
Fill in each blank
Answer:
5.In the picture, O is the circumcentre of △ABC. x =
Answer:
6.In △ABC, ∠C = 90°, AC = 21 cm and BC = 72 cm. The circumcircle's radius =
Answer:
7.G is the centroid of △ABC, BE is a median and BE = 6 cm. BG = cm, GE = cm
Fill in each blank
Answer:
8.G is the centroid of △ABC, D the middle of BC, and △ABC has an area of 18 cm². The area of △GBD =
Answer:
9.In △ABC, AD ⊥ BC with D on BC, and ∠ACB = 78°. ∠DAC =
Answer:
Part B: Choose, or true or false (2 marks each)
10.In the picture, OP bisects ∠AOB, PM ⊥ OA and PN ⊥ OB. So PM = PN ( ? ). Which reason goes in the gap?
- A. given
- B. ∠ bisector theorem
- C. ⊥ bisector theorem
- D. converse of ∠ bisector theorem
Choose the right answer
Answer:
11.A point P is 8 cm and 9 cm from the two arms of ∠AOB, measured at right angles. P lies on the bisector of ∠AOB.
True or false?
Answer:
12.In △ABC, ∠B = 50° and ∠C = 55°. In which picture is P the centroid of △ABC?
- A
- B
- C
- D
Choose the right picture
Answer:
Part C: Word problems (4 marks each)
13.Two straight roads meet at O at 44°. A lamp P stands on the bisector of that angle, 18 m from one road. P is m from the other road, and OP makes ° with a road.
Fill in each blank
Working:
Answer:
14.Two villages A and B are 180 m apart. A well P is dug on the perpendicular bisector of AB, 120 m from AB's middle M. AM = m, and the well is m from each village.
Fill in each blank
Working:
Answer:
15.A triangular lawn ABC has ∠B = 76° and ∠C = 56°. A sprinkler I must be equally far from all three edges. ∠IBC = °, ∠ICB = °, ∠BIC = °
Fill in each blank
Working:
Answer:
16.Three houses A, B and C form a right triangle with ∠C = 90°, AC = 168 m and BC = 576 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:
17.A triangular sign's centroid G is on the median AD, with GD = (x + 1) cm and AG = (3x − 6) cm. What is x?
Working:
Answer:
18.A triangular card about 39 cm across is to balance on a pencil point. Under which centre should the point go?
- A. circumcentre
- B. orthocentre
- C. incentre
- D. centroid
Choose the right answer
Working:
Answer:
Show the answers and working
Centres of a triangle worksheet: Answers and working
Part A: Working out and filling in
1. 20 cm
P is on AB's perpendicular bisector, so it is equally far from A and B: x = PA = 20 (⊥ bisector theorem).
2. 2
PA = PB (⊥ bisector theorem): 2x + 1 = 3x − 1, so x = 2. Check: PA = PB = 5 cm.
3. 4 cm
The incentre is equally far from all three sides (∠ bisector theorem): 4 cm.
4. 26, 23, 131
BI and CI are bisectors: ∠IBC = 26° and ∠ICB = 23°; ∠BIC = 180° − 26° − 23° = 131°.
5. 35°
OA = OB (the circumcentre is equally far from the vertices), so △OAB is isosceles: x = ∠OAB = 35° (base ∠s, isos. △).
6. 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.
7. 4, 2
BG : GE = 2 : 1, 3 parts of 6 ÷ 3 = 2 cm each: BG = 4 cm and GE = 2 cm.
8. 3 cm²
The three medians cut △ABC into 6 small triangles of equal area: 18 ÷ 6 = 3 cm². (Or: △ABD is half, and △GBD a third of that.)
9. 12°
In △ADC, ∠ADC = 90°: ∠DAC = 180° − 90° − 78° = 12°.
Part B: Choose, or true or false
10. B. ∠ bisector theorem
P is on the bisector, so it is equally far from the arms: ∠ bisector theorem.
11. False
Only a point equally far from both arms lies on the bisector (converse of ∠ bisector theorem).
12. A
The centroid comes from the medians: lines from the vertices to the opposite sides' middles, each side's halves ticked, no right angles.
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. 18, 22
A point on the bisector is equally far from both arms (∠ bisector theorem): 18 m; the bisector halves 44° into two of 22°.
14. 90, 150
AM = 180 ÷ 2 = 90 m; PA = 902 + 1202 = 150 m (Pyth. thm.), and PB = PA (⊥ bisector theorem).
15. 38, 28, 114
The point equally far from the sides is the incentre, on the bisectors: ∠IBC = 38°, ∠ICB = 28°, ∠BIC = 180° − 38° − 28° = 114°.
16. 600, 300
AB = 1682 + 5762 = 600 m; the point equally far from three corners is the circumcentre, at a right triangle's hypotenuse's middle: 600 ÷ 2 = 300 m.
17. 8
AG = 2 × GD: 3x − 6 = 2(x + 1), so x = 8; the median AD = 3 × 9 = 27 cm.
18. D. centroid
The centroid (where the medians meet) is the balance point: each median halves the area.
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