In △ABC, AD ⊥ BC with D on BC, and ∠ACB = 54°. ∠DAC = □

36°. In △ADC, ∠ADC = 90°: ∠DAC = 180° − 90° − 54° = 36°.

In △ABC, ∠BAC = 124°, and H is the orthocentre (outside the triangle). ∠BHC = □

56°. The heights BE and CF, carried on, meet at H; in AEHF, ∠E = ∠F = 90° and ∠EAF = 124° (vert. opp. ∠s), so ∠BHC = 360° − 90° − 90° − 124° = 56°.

In △ABC, ∠A = 25°, ∠B = 65° and ∠C = 90°. Where is its orthocentre? A. inside, at no vertex B. at the middle of AB C. outside the triangle D. at the vertex C Choose the right answer

D. at the vertex C. AC ⊥ BC, so AC and BC are both heights, meeting at C: the orthocentre is the right-angle vertex C.

Centres of a triangle · Ages 14–15

The orthocentre

Orthocentre: where the heights meet; at the right angle in a right triangle.

How to do it

Method

Orthocentre: where the heights meet; at the right angle in a right triangle

∠A = 50°: ∠ABE = 90° − 50° = 40°

Another way

∠BHC = 180° − ∠A

360° − 90° − 90° − 50° = 130°

In full

From each vertex draw the perpendicular to the opposite side (a height). A triangle's three heights (or the lines they lie on) meet at one point H, the orthocentre. Where it lies: inside an acute triangle; at the right-angle vertex of a right-angled one; outside an obtuse one, where the heights must be carried on to meet. Chasing angles: heights BE ⊥ AC and CF ⊥ AB. ∠A = 50°: in △ABE, ∠ABE = 90° − 50° = 40°. Another way: AFHE has two right angles, so ∠FHE = 360° − 90° − 90° − 50° = 130°, and ∠BHC = 130° (vert. opp. ∠s): ∠BHC = 180° − ∠A.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
In △ABC, AD ⊥ BC with D on BC, and ∠ACB = 54°. ∠DAC =
Show the answer and working
Answer: 36°
In △ADC, ∠ADC = 90°: ∠DAC = 180° − 90° − 54° = 36°.
Example 2
In △ABC, ∠BAC = 124°, and H is the orthocentre (outside the triangle). ∠BHC =
Show the answer and working
Answer: 56°
The heights BE and CF, carried on, meet at H; in AEHF, ∠E = ∠F = 90° and ∠EAF = 124° (vert. opp. ∠s), so ∠BHC = 360° − 90° − 90° − 124° = 56°.
Example 3
In △ABC, ∠A = 25°, ∠B = 65° and ∠C = 90°. Where is its orthocentre?
- A. inside, at no vertex
- B. at the middle of AB
- C. outside the triangle
- D. at the vertex C
Choose the right answer
Show the answer and working
Answer: D. at the vertex C
AC ⊥ BC, so AC and BC are both heights, meeting at C: the orthocentre is the right-angle vertex C.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.In △ABC, AD ⊥ BC with D on BC, and ∠ACB = 50°. ∠DAC =
Answer:
2.Where do which three lines meet to give a triangle's orthocentre?
- A. the three heights
- B. the three sides' perpendicular bisectors
- C. the three angle bisectors
- D. the three medians
Choose the right answer
Answer:
3.H is the orthocentre of the acute triangle ABC, where the heights BE and CF meet, and ∠BAC = 58°. ∠ABE = °, ∠ACF = °
Fill in each blank
Answer:
4.In △ABC, ∠BAC = 118°, and H is the orthocentre (outside the triangle). ∠BHC =
Answer:
5.In △ABC, ∠A = 20°, ∠B = 70° 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:

All the answers

1. 40°
2. A. the three heights
3. 32, 32
4. 62°
5. C. at the vertex C

Free worksheet: The orthocentreWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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