Question

The co-ordinates of the vertices of a triangle are (4, 6), (0, 4) and (5, 5) then the co-ordinates of the centroid of the triangle are

• A. (5, 3)
• B. (3, 4)
• C. (4, 4)
• D. (3, 5)

Hint:

The centroid is essentially the "average point" of the triangle's vertices. The formula is a straightforward extension of the midpoint formula from two points to three.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The centroid of a triangle is the point where its three medians intersect. The coordinates of the centroid are the average of the x-coordinates and the average of the y-coordinates of the three vertices.

Step 2: Key Formula or Approach:
The formula for the centroid \((x_c, y_c)\) of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is:
\[ (x_c, y_c) = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]

Step 3: Detailed Explanation:
The given vertices are \((4, 6)\), \((0, 4)\), and \((5, 5)\).
Calculate the x-coordinate of the centroid:
\[ x_c = \frac{4 + 0 + 5}{3} = \frac{9}{3} = 3 \] Calculate the y-coordinate of the centroid:
\[ y_c = \frac{6 + 4 + 5}{3} = \frac{15}{3} = 5 \] The coordinates of the centroid are \((3, 5)\).

Step 4: Final Answer:
The co-ordinates of the centroid of the triangle are (3, 5).