Triangle Centroid Problem - Detailed Solution and Interactive Quiz

Triangle Centroid Problem [IIT 1964]

Question: The centroid of a triangle, whose vertices are \( (2,\, 1) \), \( (5,\, 2) \), and \( (3,\, 4) \), is:

(a) \( \left(\frac{8}{3},\, \frac{7}{3}\right) \)

(b) \( \left(\frac{10}{3},\, \frac{7}{3}\right) \)

(c) \( \left(-\frac{10}{3},\, \frac{7}{3}\right) \)

(d) \( \left(\frac{10}{3},\, -\frac{7}{3}\right) \)

Detailed Step-by-Step Explanation

Step 1: The centroid \( G \) of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), \( (x_3, y_3) \) is given by:

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

Step 2: Identify the coordinates of the given vertices:
\( A = (2,\, 1) \), \( B = (5,\, 2) \), \( C = (3,\, 4) \)

Step 3: Substitute the coordinates into the centroid formula:

\( x\text{-coordinate} = \frac{2+5+3}{3} = \frac{10}{3} \)
\( y\text{-coordinate} = \frac{1+2+4}{3} = \frac{7}{3} \)

Therefore, the centroid is:

\( G = \left(\frac{10}{3},\, \frac{7}{3}\right) \)

This corresponds to option (b).