Using position vectors, we confirm properties of the centroid of a triangle and several geometric figures.
Properties of the Centroid
The centroid G of triangle ABC is the intersection of the three medians (segments connecting each vertex to the midpoint of the opposite side), and it internally divides each median in the ratio 2:1.
→AG : →GM = 2 : 1 (M is the midpoint of BC)
Example ①
Find the centroid G of A(0, 0), B(6, 0), and C(0, 6). Also verify AG:GM (M is the midpoint of BC).
Solution: G=(2, 2). M=((6+0)/2, (0+6)/2)=(3, 3).
→AG=(2, 2)、→GM=(1, 1). Therefore, →AG=2→GM、AG:GM=2:1。
Find the centroid G of A(0, 0), B(6, 0), and C(0, 6). Also verify AG:GM (M is the midpoint of BC).
Solution: G=(2, 2). M=((6+0)/2, (0+6)/2)=(3, 3).
→AG=(2, 2)、→GM=(1, 1). Therefore, →AG=2→GM、AG:GM=2:1。
Working with Figures Using Position Vectors
Four points A, B, C, and D form a parallelogram ⇔ →AB = →DC (or →a + →c = →b + →d).
Example ②
For A(1, 2), B(4, 3), and C(6, 7), find D such that ABCD is a parallelogram.
Solution: →AB = →DC → →OD = →OC − →AB = (6−3, 7−1) = D(3, 6)
For A(1, 2), B(4, 3), and C(6, 7), find D such that ABCD is a parallelogram.
Solution: →AB = →DC → →OD = →OC − →AB = (6−3, 7−1) = D(3, 6)
Key Points
- The centroid divides each median in the ratio 2:1 (2 on the vertex side and 1 on the midpoint side)
- Position vector of the centroid: (→a + →b + →c)/3
- Parallelogram ABCD ⇔ →AB = →DC
Practice Problems
- Find the centroid of A(2, 1), B(5, 7), and C(−1, 4).
- For A(0, 0), B(4, 1), and C(5, 5), find D such that ABCD is a parallelogram.
- For the centroid G of triangle ABC, calculate →AG + →BG + →CG.
Answers and Explanations
- Answer: (2, 4)
Explanation: (2+5−1)/3=2、(1+7+4)/3=4。 - Answer: D(1, 4)
Explanation: →AB=(4,1)、→DC=→AB → →OD=→OC−→AB=(5−4, 5−1)=(1, 4)。 - Answer: →0
Explanation: →AG=(→a+→b+→c)/3−→a. Simplifying similarly, the sum of the three vectors is →0.