In coordinate geometry, a triangle has vertices A(2, 5), B(4, 4), and C(7, 3). Using the formula for the centroid, find the coordinates of the centroid of triangle ABC.

Introduction / Context:
This coordinate geometry question asks you to find the centroid of a triangle when the coordinates of its vertices are given. The centroid is the point where the three medians of a triangle intersect and is sometimes called the center of mass for a triangle with uniform density. Knowing the centroid formula is very useful in analytic geometry and applications such as physics and computer graphics.

Given Data / Assumptions:

• Vertex A has coordinates (2, 5).
• Vertex B has coordinates (4, 4).
• Vertex C has coordinates (7, 3).
• We must find the coordinates of the centroid G of triangle ABC.
• The coordinate system is a standard Cartesian plane.

Concept / Approach:
The centroid G of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is given by the simple averaging formulas: Gx = (x1 + x2 + x3) / 3 and Gy = (y1 + y2 + y3) / 3. That is, the centroid is the arithmetic mean of the x-coordinates and the arithmetic mean of the y-coordinates of the vertices. We simply substitute the given coordinates of A, B, and C into these formulas and simplify to get the centroid.

Step-by-Step Solution:
1) Label the coordinates: A(2, 5), B(4, 4), and C(7, 3). 2) Compute the sum of the x-coordinates: x1 + x2 + x3 = 2 + 4 + 7 = 13. 3) Compute the sum of the y-coordinates: y1 + y2 + y3 = 5 + 4 + 3 = 12. 4) Apply the centroid formula for the x-coordinate: Gx = (x1 + x2 + x3) / 3 = 13 / 3. 5) Apply the centroid formula for the y-coordinate: Gy = (y1 + y2 + y3) / 3 = 12 / 3 = 4. 6) Therefore the centroid G has coordinates (13/3, 4).

Verification / Alternative check:
To build intuition, note that the centroid always lies inside the triangle and each median divides the triangle into two smaller triangles of equal area. The vertex coordinates here are fairly close to each other, so it is reasonable that the centroid x-coordinate 13/3 is between 2 and 7 and the y-coordinate 4 is between 3 and 5. Plotting the points roughly on a graph would show the centroid located in the interior, roughly central, which confirms that the computed point is plausible.

Why Other Options Are Wrong:
Option b (4, 13/3) swaps the roles of the x and y coordinates. Option c (3, 4) uses an incorrect average for the x-coordinates. Option d (13/3, 12/3) writes the y-coordinate as 12/3 explicitly but does not match the simpler 4 in the option style, and it may be a distractor for careless simplification. Option e (5, 4) takes a middle x-value by eye rather than the correct average. Only option a accurately applies the centroid formula in both coordinates.

Common Pitfalls:
A common mistake is to divide only by 2, as if averaging two points instead of three, or to average only the x-coordinates and forget to average the y-coordinates. Others might mis add one of the coordinates, for example 2 + 4 + 7 incorrectly. Writing the formula clearly and then substituting step by step ensures accuracy. Double checking the arithmetic, especially the division by 3, helps avoid small but important errors.

Final Answer:
The centroid of triangle ABC with vertices A(2, 5), B(4, 4), and C(7, 3) is the point (13/3, 4).