(vi) Find the coordinates of the centroid of triangle ABC with vertices

General
1

(vi) Find the coordinates of the centroid of triangle ABC with vertices A(2,4), B(6,1), and C(8,5). (a) (15.5,10/3) (b) (3,0) (c) (16/3,3) (d) (16/3,10/3)

2

How to Find the Centroid of Triangle ABC with Given Vertices

When you’re tasked with finding the centroid of a triangle, you’re essentially looking for the “geometric center” or “balance point” of the triangle. The centroid is calculated by taking the average of the x-coordinates and the average of the y-coordinates of the triangle’s vertices.

Vertices of Triangle ABC

The problem gives you the vertices of triangle ABC:

  • A: (2, 4)
  • B: (6, 1)
  • C: (8, 5)

Formula for the Centroid

The formula to calculate the centroid (G(x, y)) for a triangle with vertices (A(x_1, y_1)), (B(x_2, y_2)), and (C(x_3, y_3)) is:

[
G\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)
]

Applying the Formula

Step 1: Calculate the x-coordinate of the centroid:

  • Sum of the x-coordinates: x_1 + x_2 + x_3 = 2 + 6 + 8 = 16
  • Average the sum by dividing by 3: \frac{16}{3}

Step 2: Calculate the y-coordinate of the centroid:

  • Sum of the y-coordinates: y_1 + y_2 + y_3 = 4 + 1 + 5 = 10
  • Average the sum by dividing by 3: \frac{10}{3}

Centroid Coordinates

Thus, the coordinates of the centroid (G) are:

[
G\left(\frac{16}{3}, \frac{10}{3}\right)
]

This matches option (d), which is (16/3, 10/3).

Why is the Centroid Important?

Understanding the centroid is crucial in many fields such as physics, engineering, and computer graphics because it helps determine the center of mass of an object, predict behaviour under certain conditions, and balance structures.

Verifying Other Options

Let’s briefly verify why other options do not fit:

  • (a) (15.5, 10/3): Incorrect as neither math nor calculation fits the formula for centroid.
  • (b) (3, 0): Incorrect both x and y values do not match the calculated ones.
  • (c) (16/3, 3): The x-value is correct, but the y-value doesn’t match.

Conclusion

By following the correct and systematic method of averaging the coordinates, we can ensure that option (d) (16/3, 10/3) is the accurate choice for the centroid of triangle ABC. Employing this method universally provides valuable insight across various practical applications. @LectureNotes