Geometry and proof · Ages 14–15
The orthocentre worksheet
Part of: Centres of a triangle
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The orthocentre 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 △ABC, AD ⊥ BC with D on BC, and ∠ACB = 74°. ∠DAC =
Answer:
2.In the picture, H is the orthocentre of △ABC, and ∠ACB = 60°. x =
Answer:
3.In △ABC, AD ⊥ BC with D on BC, and ∠ACB = 68°. ∠DAC =
Answer:
4.H is the orthocentre of the acute triangle ABC, and ∠BAC = 78°. ∠BHC =
Answer:
5.H is the orthocentre of the acute triangle ABC, where the heights BE and CF meet, and ∠BAC = 72°. ∠ABE = °, ∠ACF = °
Fill in each blank
Answer:
6.H is the orthocentre of the acute triangle ABC, and ∠BAC = 34°. ∠BHC =
Answer:
7.H is the orthocentre of the acute triangle ABC, where the heights BE and CF meet, and ∠BAC = 42°. ∠ABE = °, ∠ACF = °
Fill in each blank
Answer:
8.In △ABC, ∠BAC = 126°, and H is the orthocentre (outside the triangle). ∠BHC =
Answer:
9.In △ABC, ∠BAC = 134°, and H is the orthocentre (outside the triangle). ∠BHC =
Answer:
Part B: Choose, or true or false (2 marks each)
10.Where do which three lines meet to give a triangle's orthocentre?
- A. the three angle bisectors
- B. the three heights
- C. the three medians
- D. the three sides' perpendicular bisectors
Choose the right answer
Answer:
11.In △ABC, ∠A = 30°, ∠B = 60° and ∠C = 90°. Where is its orthocentre?
- A. at the middle of AB
- B. outside the triangle
- C. at the vertex C
- D. inside, at no vertex
Choose the right answer
Answer:
12.In △ABC, ∠A = 45°, ∠B = 45° and ∠C = 90°. Where is its orthocentre?
- A. inside, at no vertex
- B. outside the triangle
- C. at the vertex C
- D. at the middle of AB
Choose the right answer
Answer:
Part C: Word problems (4 marks each)
13.An obtuse triangular bracket ABC has ∠A = 116°. The heights from B and C, carried on, meet at H outside it. What is ∠BHC?
Working:
Answer:
14.A triangular sign ABC has ∠B = 60° and ∠C = 64°, with a string AD hanging from A at right angles to BC. ∠BAD = °, ∠DAC = °
Fill in each blank
Working:
Answer:
15.A right-triangular roof frame ABC has ∠C = 90°, AC = 9 m, BC = 12 m and AB = 15 m. Its three heights meet at the orthocentre H. How far is H from AB?
Working:
Answer:
16.An acute triangular tent frame ABC has ∠A = 76°. Struts BE ⊥ AC and CF ⊥ AB meet at H. ∠ABE = °, ∠BHC = °
Fill in each blank
Working:
Answer:
17.A triangular sign ABC has ∠B = 68° and ∠C = 78°, with a string AD hanging from A at right angles to BC. ∠BAD = °, ∠DAC = °
Fill in each blank
Working:
Answer:
18.A right-triangular roof frame ABC has ∠C = 90°, AC = 24 m, BC = 18 m and AB = 30 m. Its three heights meet at the orthocentre H. How far is H from AB?
Working:
Answer:
Show the answers and working
The orthocentre worksheet: Answers and working
Part A: Working out and filling in
1. 16°
In △ADC, ∠ADC = 90°: ∠DAC = 180° − 90° − 74° = 16°.
2. 60°
BE ⊥ AC, so in △BEC, ∠HBD = 90° − 60° = 30°; AD ⊥ BC, so in △BHD, x = 180° − 90° − 30° = 60°.
3. 22°
In △ADC, ∠ADC = 90°: ∠DAC = 180° − 90° − 68° = 22°.
4. 102°
The heights BE and CF meet AC and AB at right angles, so in AFHE, ∠FHE = 360° − 90° − 90° − 78° = 102°; ∠BHC = ∠FHE (vert. opp. ∠s) = 102°.
5. 18, 18
In △ABE, ∠AEB = 90°: ∠ABE = 90° − 72° = 18°; likewise in △ACF, ∠ACF = 18°.
6. 146°
The heights BE and CF meet AC and AB at right angles, so in AFHE, ∠FHE = 360° − 90° − 90° − 34° = 146°; ∠BHC = ∠FHE (vert. opp. ∠s) = 146°.
7. 48, 48
In △ABE, ∠AEB = 90°: ∠ABE = 90° − 42° = 48°; likewise in △ACF, ∠ACF = 48°.
8. 54°
The heights BE and CF, carried on, meet at H; in AEHF, ∠E = ∠F = 90° and ∠EAF = 126° (vert. opp. ∠s), so ∠BHC = 360° − 90° − 90° − 126° = 54°.
9. 46°
The heights BE and CF, carried on, meet at H; in AEHF, ∠E = ∠F = 90° and ∠EAF = 134° (vert. opp. ∠s), so ∠BHC = 360° − 90° − 90° − 134° = 46°.
Part B: Choose, or true or false
10. B. the three heights
The orthocentre is where the heights (perpendiculars from the vertices to the opposite sides) meet.
11. C. at the vertex C
AC ⊥ BC, so AC and BC are both heights, meeting at C: the orthocentre is the right-angle vertex C.
12. C. at the vertex C
AC ⊥ BC, so AC and BC are both heights, meeting at C: the orthocentre is the right-angle vertex C.
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. 64°
AEHF has two right angles and ∠EAF = 116° (vert. opp. ∠s): ∠BHC = 360° − 90° − 90° − 116° = 64°.
14. 30, 26
AD is a height: in △ABD, ∠BAD = 90° − 60° = 30°; in △ADC, ∠DAC = 90° − 64° = 26°.
15. 7.2 m
A right triangle's orthocentre is the right-angle vertex C; C's height to AB = 9 × 12 ÷ 15 = 7.2 m (the area worked out two ways).
16. 14, 104
In △ABE, ∠ABE = 90° − 76° = 14°; in AFHE, ∠FHE = 360° − 90° − 90° − 76° = 104° = ∠BHC (vert. opp. ∠s).
17. 22, 12
AD is a height: in △ABD, ∠BAD = 90° − 68° = 22°; in △ADC, ∠DAC = 90° − 78° = 12°.
18. 14.4 m
A right triangle's orthocentre is the right-angle vertex C; C's height to AB = 24 × 18 ÷ 30 = 14.4 m (the area worked out two ways).
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