Finding the Centroid Coordinates of a Triangle

The centroid of a triangle is a very important point. It is the point where the three medians of the triangle intersect. A median is a line segment drawn from a vertex to the midpoint of the opposite side.

A key property of the centroid is that it divides each median in a 2:1 ratio, with the longer segment being towards the vertex.

Understanding the Centroid and Median Property

Let's consider a triangle ABC. Let A be one vertex and D be the midpoint of the opposite side BC. The line segment AD is a median. If G is the centroid of the triangle, then G lies on the median AD such that the distance AG is twice the distance GD. Mathematically, this ratio is expressed as AG : GD = 2 : 1.

Using the Section Formula to Find Centroid Coordinates

Since the centroid G divides the median AD in the ratio 2:1, we can use the section formula to find its coordinates. The section formula helps us find the coordinates of a point that divides a line segment connecting two points in a given ratio.

Let the vertex A be \((x_1, y_1)\) and the midpoint D be \((x_2, y_2)\). The centroid G \((x, y)\) divides the segment AD in the ratio \(m:n = 2:1\).

The section formula for the coordinates of G is:

\(x = \frac{m x_2 + n x_1}{m+n}\)

\(y = \frac{m y_2 + n y_1}{m+n}\)

Applying the Formula to the Given Triangle

In this question, we are given:

• Vertex A = (2, -3). So, \((x_1, y_1) = (2, -3)\).
• Midpoint of BC, D = (2, 4). So, \((x_2, y_2) = (2, 4)\).
• The centroid G divides AD in the ratio 2:1. So, \(m = 2\) and \(n = 1\).

Now, let's calculate the coordinates of the centroid G \((x, y)\) using the section formula:

Calculating the x-coordinate of the Centroid

\(x = \frac{2(2) + 1(2)}{2+1}\)

\(x = \frac{4 + 2}{3}\)

\(x = \frac{6}{3}\)

\(x = 2\)

Calculating the y-coordinate of the Centroid

\(y = \frac{2(4) + 1(-3)}{2+1}\)

\(y = \frac{8 - 3}{3}\)

\(y = \frac{5}{3}\)

So, the coordinates of the centroid G are \((2, \frac{5}{3})\).

Summarizing the Centroid Coordinates

Based on our calculations, the centroid of triangle ABC is located at the point (2, 5/3).

Let's check the given options:

• Option 1: (1, 0)
• Option 2: (0, -5/2)
• Option 3: (2, 5)
• Option 4: (2, 5/3)

Our calculated coordinates (2, 5/3) match Option 4.

Revision Table: Centroid Formulas

Additional Information: Properties of Triangle Centroid

• The centroid is always inside the triangle.
• It is the triangle's center of mass. If the triangle were a physical object of uniform density, it would balance at the centroid.
• The centroid divides the triangle into six smaller triangles of equal area.
• For a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\), the coordinates of the centroid G are simply the average of the coordinates: \(G = \left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)\). This is another common formula for finding the centroid coordinates. The problem here used the median property, which is related.

This method using the section formula with a vertex and the midpoint of the opposite side is particularly useful when the coordinates of all three vertices are not directly given, but a vertex and a midpoint are.