More about the centroid
Part 1
1. Construct TWO of the medians for the triangle shown below to find the centroid for the triangle.
2. In what ratio does the centroid cut the medians?
Part 2
Drag any of the vertices of the triangle shown below.
a. What do you notice?
b. What generalisation can you make?
Part 3 - A proof
In the diagram below;
i. G is the centroid of triangle ABC.
ii. DE and GH are parallel to AB.
Hint: for questions 1, 2 and 3 below it is better to maintain the situation where there is a right-angle at B and BC is parallel to the x-axis.
1. Explain the values of the ratios shown in the diagram below. Hint: refer to your results from Part 1 above.
2. A has coordinates (x1, y1), B has coordinates (x2, y2) and C has coordinates (x3, y3). Use this information and the diagram below to answer the following.
i. What is the x-coordinate of the point E?
ii. What is the distance from B to the point E?
iii. What is the distance from B to point H?
iv. Hence, show that the x-coordinate of the point G is the average of the x-coordinates of the vertices of triangle ABC.
3. By similar logic as that above, show that the y-coordinate of the point G is the average of the y-coordinates of the vertices of triangle ABC.
Hint: you can construct additional lines on the diagram if needed.
4. What happens when the orientation and shape of the triangle is changed and does it make the result harder to prove?