Geometry and proof · Ages 14–15

The centroid worksheet

Part of: Centres of a triangle

In the print window, choose “Save as PDF” to download it. The answers print on the last page.

Geometry and proof · Ages 14–15

Numsprite
numsprite.com

Name:Date:Score:／48

Write each answer in the space. For word problems, show your working and write a sentence answer.

1.In the picture, G is the centroid of △ABC, and the median AD = 30 cm. AG =
Answer:
2.In the picture, G is the centroid of △ABC, and the median AD = 15 cm. GD =
Answer:
3.In the picture, G is the centroid of △ABC, and the median AD = 9 cm. GD =
Answer:
4.In the picture, G is the centroid of △ABC, and the median AD = 27 cm. AG =
Answer:
5.G is the centroid of △ABC, BE is a median and BE = 12 cm. BG = cm, GE = cm
Fill in each blank
Answer:
6.G is the centroid of △ABC, AD is a median and AG = 18 cm. GD =
Answer:
7.G is the centroid of △ABC, BE is a median and BE = 15 cm. BG = cm, GE = cm
Fill in each blank
Answer:
8.G is the centroid of △ABC, AD is a median, AG = (x + 4) cm and GD = (x − 1) cm. x =
Answer:
9.G is the centroid of △ABC, D the middle of BC, and △ABC has an area of 36 cm². The area of △GBD =
Answer:
10.G is the centroid of △ABC, D the middle of BC, and △ABC has an area of 120 cm². The area of △GBD =
Answer:
11.G is the centroid of △ABC, AD is a median, AG = (x + 3) cm and GD = (x − 3) cm. x =
Answer:

13.A triangular sign's centroid G is on the median AD, with GD = (x + 5) cm and AG = (3x − 1) cm. What is x?
Working:
Answer:
14.A triangular card balances on a pencil point at its centroid G. One median, AD, is 27 cm. G is cm from the corner A and cm from D, the middle of BC.
Fill in each blank
Working:
Answer:
15.A smith finds a triangular plate's balance point G, 4 cm from the corner A on the median from A to the opposite side's middle. How long is that median?
Working:
Answer:
16.A triangular piece of cloth of 210 cm² is cut along its three medians. Each small piece is cm²; △ABG (G the centroid) is cm².
Fill in each blank
Working:
Answer:
17.A triangular card balances on a pencil point at its centroid G. One median, AD, is 21 cm. G is cm from the corner A and cm from D, the middle of BC.
Fill in each blank
Working:
Answer:
18.A triangular sign's centroid G is on the median AD, with GD = (x + 1) cm and AG = (3x − 3) cm. What is x?
Working:
Answer:

Show the answers and working

The centroid worksheet: Answers and working

1. 20 cm
The centroid cuts a median 2 : 1: x = AG = 30 × 23 = 20.
2. 5 cm
AG : GD = 2 : 1, so GD is a third of the median: x = 15 ÷ 3 = 5.
3. 3 cm
AG : GD = 2 : 1, so GD is a third of the median: x = 9 ÷ 3 = 3.
4. 18 cm
The centroid cuts a median 2 : 1: x = AG = 27 × 23 = 18.
5. 8, 4
BG : GE = 2 : 1, 3 parts of 12 ÷ 3 = 4 cm each: BG = 8 cm and GE = 4 cm.
6. 9 cm
AG : GD = 2 : 1: GD = 18 ÷ 2 = 9 cm.
7. 10, 5
BG : GE = 2 : 1, 3 parts of 15 ÷ 3 = 5 cm each: BG = 10 cm and GE = 5 cm.
8. 6
AG = 2 × GD: x + 4 = 2(x − 1), so x = 6. Check: AG = 10 cm is twice GD.
9. 6 cm²
The three medians cut △ABC into 6 small triangles of equal area: 36 ÷ 6 = 6 cm². (Or: △ABD is half, and △GBD a third of that.)
10. 20 cm²
The three medians cut △ABC into 6 small triangles of equal area: 120 ÷ 6 = 20 cm². (Or: △ABD is half, and △GBD a third of that.)
11. 9
AG = 2 × GD: x + 3 = 2(x − 3), so x = 9. Check: AG = 12 cm is twice GD.

12. A. the three medians
The centroid is where the medians (vertex to the opposite side's middle) meet.

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

13. 11
AG = 2 × GD: 3x − 1 = 2(x + 5), so x = 11; the median AD = 3 × 16 = 48 cm.
14. 18, 9
The centroid cuts a median 2 : 1: AG = 27 × 23 = 18 cm and GD = 27 ÷ 3 = 9 cm.
15. 6 cm
The balance point is the centroid, and AG is 23 of the median: median = 4 ÷ 2 × 3 = 6 cm.
16. 35, 70
The three medians cut it into 6 triangles of equal area: 210 ÷ 6 = 35 cm²; △ABG is 2 of them: 70 cm².
17. 14, 7
The centroid cuts a median 2 : 1: AG = 21 × 23 = 14 cm and GD = 21 ÷ 3 = 7 cm.
18. 5
AG = 2 × GD: 3x − 3 = 2(x + 1), so x = 5; the median AD = 3 × 6 = 18 cm.

Numsprite · numsprite.com/en/worksheets/centres-of-a-triangle/centroid-of-a-triangle