What are the coordinates of the centroid of a triangle whose vertices are A(2, 5), B(-4, 0) and C(5, 4)?

Introduction / Context:
This problem is from coordinate geometry and asks you to find the centroid of a triangle from the coordinates of its vertices. The centroid is the point where the three medians of the triangle intersect and has a simple formula in terms of the coordinates of the vertices.

Given Data / Assumptions:

• Triangle vertices are A(2, 5), B(-4, 0) and C(5, 4).
• We need the centroid coordinates G(x, y).
• The centroid is the average of the coordinates of the three vertices.

Concept / Approach:
For any triangle with vertices (x1, y1), (x2, y2) and (x3, y3), the centroid G has coordinates G = ((x1 + x2 + x3) / 3, (y1 + y2 + y3) / 3). We simply plug in the given coordinates and simplify to get the required point.

Step-by-Step Solution:
Step 1: Identify the coordinates of the vertices: A(2, 5), B(-4, 0), C(5, 4).Step 2: Compute the x coordinate of the centroid: (2 + (-4) + 5) / 3 = (2 - 4 + 5) / 3 = 3 / 3 = 1.Step 3: Compute the y coordinate of the centroid: (5 + 0 + 4) / 3 = 9 / 3 = 3.Step 4: Therefore the centroid G is at (1, 3).

Verification / Alternative check:
You can check that the centroid lies roughly at the middle of the triangle by plotting the three vertices on a rough sketch and visually estimating the intersection point of medians, which should be close to (1, 3), supporting the calculation.

Why Other Options Are Wrong:
The points (-1, 3), (1, -3), (-1, -3) and (0, 3) do not satisfy the centroid formula for the given vertices. They usually arise from sign mistakes or averaging only two points instead of all three.

Common Pitfalls:
Typical errors include forgetting to divide by 3, mixing x and y coordinates, or incorrectly adding negative values. Carefully grouping the sums and dividing by 3 for each coordinate prevents these mistakes.

Final Answer:
The coordinates of the centroid are (1, 3).