Centres of a triangle · Ages 14–15
The centroid
Centroid: where the medians meet, AG : GD = 2 : 1.
How to do it
Method
Centroid: where the medians meet, AG : GD = 2 : 1
AD = 12: AG = 12 × 23 = 8, GD = 4
Another way
Three parts
12 ÷ 3 = 4, AG = 2 × 4 = 8
In full
A segment joining a vertex to the middle of the opposite side is a median. A triangle's three medians meet at one point G, the centroid. It cuts each median 2 : 1, the part at the vertex twice the part at the mid-point: AG : GD = 2 : 1, so AG is 23 of the median and GD is 13. A median AD of 12 cm gives AG = 12 × 23 = 8 cm and GD = 4 cm. Another way: by parts. AG is 2 parts and GD 1, 3 in all, each 12 ÷ 3 = 4 cm. The centroid is always inside; a card triangle balances on it. Each median cuts the triangle into two halves of equal area.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
In the picture, G is the centroid of △ABC, and the median AD = 30 cm. GD =
Show the answer and working
Answer: 10 cm
AG : GD = 2 : 1, so GD is a third of the median: x = 30 ÷ 3 = 10.
Example 2
Where do which three lines meet to give a triangle's centroid?
- A. the three medians
- B. the three heights
- C. the three angle bisectors
- D. the three sides' perpendicular bisectors
Choose the right answer
Show the answer and working
Answer: A. the three medians
The centroid is where the medians (vertex to the opposite side's middle) meet.
Example 3
G is the centroid of △ABC, D the middle of BC, and △ABC has an area of 12 cm². The area of △GBD =
Show the answer and working
Answer: 2 cm²
The three medians cut △ABC into 6 small triangles of equal area: 12 ÷ 6 = 2 cm². (Or: △ABD is half, and △GBD a third of that.)
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.In the picture, G is the centroid of △ABC, and the median AD = 15 cm. GD =
Answer:
2.In the picture, G is the centroid of △ABC, and the median AD = 21 cm. AG =
Answer:
3.Where do which three lines meet to give a triangle's centroid?
- A. the three angle bisectors
- B. the three heights
- C. the three medians
- D. the three sides' perpendicular bisectors
Choose the right answer
Answer:
4.G is the centroid of △ABC, BE is a median and BE = 21 cm. BG = cm, GE = cm
Fill in each blank
Answer:
5.G is the centroid of △ABC, D the middle of BC, and △ABC has an area of 48 cm². The area of △GBD =
Answer:
All the answers
- 1. 5 cm
- 2. 14 cm
- 3. C. the three medians
- 4. 14, 7
- 5. 8 cm²
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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