Geometry and proof · Ages 14–15

The incentre 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.I is the incentre of △ABC, 9 cm from BC. Its distance from AB =
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2.In the picture, I is the incentre of △ABC. x =
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3.I is the incentre of △ABC, 6 cm from BC. Its distance from AB =
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4.In the picture, I is the incentre of △ABC. x =
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5.I is the incentre of △ABC, with ∠ABC = 96° and ∠ACB = 52°. ∠IBC = °, ∠ICB = °, ∠BIC = °
Fill in each blank
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6.I is the incentre of △ABC, with ∠ABC = 66° and ∠ACB = 76°. ∠IBC = °, ∠ICB = °, ∠BIC = °
Fill in each blank
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7.I is the incentre of △ABC, with ∠ABC = 70° and ∠ACB = 60°. ∠IBC = °, ∠ICB = °, ∠BIC = °
Fill in each blank
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8.In △ABC, ∠C = 90°, AC = 3 cm, BC = 4 cm and AB = 5 cm. The incircle's radius =
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9.I is the incentre of △ABC, and ∠BAC = 96°. ∠BIC =
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10.I is the incentre of △ABC, and ∠BAC = 58°. ∠BIC =
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11.In △ABC, ∠C = 90°, AC = 14 cm, BC = 48 cm and AB = 50 cm. The incircle's radius =
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13.A triangular flower bed ABC has ∠A = 100°, and a sprinkler sits at its incentre I. ∠IAB = °, ∠BIC = °
Fill in each blank
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14.A triangular lawn ABC has ∠B = 48° and ∠C = 74°. A sprinkler I must be equally far from all three edges. ∠IBC = °, ∠ICB = °, ∠BIC = °
Fill in each blank
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15.A right-triangular park has legs of 12 m and 16 m and a hypotenuse of 20 m. The largest round pond touching all three sides is dug. What is its radius?
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16.A triangular board has a perimeter of 32 cm, and its incentre I is 6 cm from every side. What is its area?
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17.A triangular lawn ABC has ∠B = 64° and ∠C = 40°. A sprinkler I must be equally far from all three edges. ∠IBC = °, ∠ICB = °, ∠BIC = °
Fill in each blank
Working:
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18.A triangular board has a perimeter of 40 cm, and its incentre I is 9 cm from every side. What is its area?
Working:
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Show the answers and working

The incentre worksheet: Answers and working

1. 9 cm
The incentre is equally far from all three sides (∠ bisector theorem): 9 cm.
2. 31°
The incentre lies on the angle bisectors, so BI bisects ∠ABC: x = ∠IBC = 31°.
3. 6 cm
The incentre is equally far from all three sides (∠ bisector theorem): 6 cm.
4. 123°
In △IBC: x = 180° − 28° − 29° = 123° (∠ sum of △).
5. 48, 26, 106
BI and CI are bisectors: ∠IBC = 48° and ∠ICB = 26°; ∠BIC = 180° − 48° − 26° = 106°.
6. 33, 38, 109
BI and CI are bisectors: ∠IBC = 33° and ∠ICB = 38°; ∠BIC = 180° − 33° − 38° = 109°.
7. 35, 30, 115
BI and CI are bisectors: ∠IBC = 35° and ∠ICB = 30°; ∠BIC = 180° − 35° − 30° = 115°.
8. 1 cm
Area = 3 × 4 ÷ 2; the area is also r × perimeter ÷ 2 (three triangles from I): r = 3 × 4 ÷ (3 + 4 + 5) = 1 cm.
9. 138°
∠ABC + ∠ACB = 180° − 96°, and ∠IBC + ∠ICB is half that; ∠BIC = 180° − (180° − 96°) ÷ 2 = 90° + 96° ÷ 2 = 138°.
10. 119°
∠ABC + ∠ACB = 180° − 58°, and ∠IBC + ∠ICB is half that; ∠BIC = 180° − (180° − 58°) ÷ 2 = 90° + 58° ÷ 2 = 119°.
11. 6 cm
Area = 14 × 48 ÷ 2; the area is also r × perimeter ÷ 2 (three triangles from I): r = 14 × 48 ÷ (14 + 48 + 50) = 6 cm.

12. C. the three angle bisectors
The incentre is where the angle bisectors meet, equally far from the three sides.

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

13. 50, 140
AI bisects ∠A: 50°; ∠BIC = 180° − (180° − 100°) ÷ 2 = 140°.
14. 24, 37, 119
The point equally far from the sides is the incentre, on the bisectors: ∠IBC = 24°, ∠ICB = 37°, ∠BIC = 180° − 24° − 37° = 119°.
15. 4 m
Area = 12 × 16 ÷ 2 = r × perimeter ÷ 2, so r = 12 × 16 ÷ (12 + 16 + 20) = 4 m; a right triangle also gives r = (12 + 16 − 20) ÷ 2.
16. 96 cm²
IA, IB and IC cut it into three triangles, each 6 cm high: area = 6 × (the three sides) ÷ 2 = 6 × 32 ÷ 2 = 96 cm².
17. 32, 20, 128
The point equally far from the sides is the incentre, on the bisectors: ∠IBC = 32°, ∠ICB = 20°, ∠BIC = 180° − 32° − 20° = 128°.
18. 180 cm²
IA, IB and IC cut it into three triangles, each 9 cm high: area = 9 × (the three sides) ÷ 2 = 9 × 40 ÷ 2 = 180 cm².

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