Three friends are playing football on a triangular field. They kick the ball…
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Three friends are playing football on a triangular field. They kick the ball to a point such that its position becomes the centroid of the field. If the coordinates of the corners of the field are (3, 1), (2, 3) and (-2, 2), find the coordinates of the centroid.
- A.
(4, 5)
- B.
(1, 2)
- C.
(5, 7)
- D.
(2, 4)
Attempted by 1 students.
Show answer & explanation
Correct answer: B
Concept
The centroid of a triangle is the point where its three medians meet — it is the average position of the three vertices, and physically it is the balance point (centre of mass) of the triangular region.
For vertices (x1, y1), (x2, y2), (x3, y3): X = (x1 + x2 + x3) / 3 and Y = (y1 + y2 + y3) / 3.
Application
Given vertices: (3, 1), (2, 3) and (-2, 2).
X-coordinate of centroid = (3 + 2 + (-2)) / 3 = 3/3 = 1.
Y-coordinate of centroid = (1 + 3 + 2) / 3 = 6/3 = 2.
So the centroid of the triangular field is (1, 2).
Cross-check
A key property of the centroid is that the position vectors from it to the three vertices sum to zero. Taking (1, 2) as the centroid: (3-1, 1-2) + (2-1, 3-2) + (-2-1, 2-2) = (2, -1) + (1, 1) + (-3, 0) = (0, 0), confirming the centroid is correctly located at (1, 2).