Exploring the Coordinates of a Triangle's Vertex Using the Centroid Formula

In this comprehensive guide, we will delve into a detailed discussion on how to determine the coordinates of a triangle's vertex using the centroid formula. Specifically, we will solve a problem where the coordinates of the centroid and two vertices of a triangle are given. We will walk you through the steps to find the coordinates of the third vertex.

What is the Centroid of a Triangle?

The centroid of a triangle is a key point that intersects the medians of the triangle. It is the point where all three medians (lines from a vertex to the midpoint of the opposite side) converge. The centroid divides each median into a ratio of 2:1.

Problem Statement

We are given the coordinates of the centroid of a triangle ABC as G (2, 2). The coordinates of vertices A and B are given as A (7, -1) and B (1, 2), respectively. Our goal is to find the coordinates of vertex C.

Using the Centroid Formula

The formula to find the coordinates of the centroid is given by:

Gx_c y_c ? [x1x2x3/3 y1y2y3/3]

Given:

Gx_c y_c (2, 2) Ax1 y1 (7, -1) Bx2 y2 (1, 2) Let the coordinates of vertex C be x3 y3.

Solving for C

From the centroid formula, we can set up the equations:

For x-coordinates:

2 (7 1 x3)/3

For y-coordinates:

2 (-1 2 y3)/3

Solving for x3:

Starting with the x-coordinate equation:

2 (8 x3)/3

Multiplying both sides by 3:

8 x3 6

Subtracting 8 from both sides:

x3 -2

Solving for y3:

Now for the y-coordinate equation:

2 (1 y3)/3

Multiplying both sides by 3:

1 y3 6

Subtracting 1 from both sides:

y3 5

Conclusion:

The coordinates of vertex C are:

C (-2, 5)

Verification Using the Median Property

Let's verify the result using the property of the centroid. The centroid divides the medians in a 2:1 ratio. If we find the midpoint E of side AB, and then use the section formula to find the coordinates of vertex C, we should get the same result.

Step-by-Step Verification

1. Find the Midpoint E of AB:

`x' (7 1)/2 4

`y' (-1 2)/2 1/2

2. Use the Section Formula to Find Coordinates of C:

`2 (2*7 x3)/3 rArr; x3 -2`

`2 (2*(-1) y3)/3 rArr; y3 5`

Therefore, the coordinates of vertex C are:

C(-2, 5)

Conclusion

By applying the centroid formula and verifying the result using the properties of medians, we have successfully determined the coordinates of vertex C for the given triangle. This method can be applied to other similar problems to find the coordinates of any vertex based on the given centroid and other vertices.