The centroid of triangle ABC has coordinates (1, -4). If the coordinates of vertices A and B are A(3, -4) and B(0, 5) respectively, what are the coordinates of vertex C?

Introduction / Context:
This coordinate geometry question asks you to find the coordinates of the third vertex of a triangle when the centroid and two vertices are known. The centroid is a special point at which the three medians of a triangle intersect, and it has a simple formula in terms of the vertices coordinates.

Given Data / Assumptions:

• Centroid G of triangle ABC is at (1, -4).
• Vertex A is at (3, -4).
• Vertex B is at (0, 5).
• Vertex C has coordinates (x, y), which are unknown.

Concept / Approach:
The centroid formula for a triangle with vertices A(x1, y1), B(x2, y2) and C(x3, y3) is:
G = ((x1 + x2 + x3) / 3, (y1 + y2 + y3) / 3).Given G and two vertices, we can set up equations for the x and y coordinates and solve for x3 and y3, which correspond to the coordinates of vertex C.

Step-by-Step Solution:
Step 1: Let C have coordinates (x, y).Step 2: Apply the centroid formula to the x coordinates: (x_A + x_B + x_C) / 3 = 1.Step 3: Substitute values: (3 + 0 + x) / 3 = 1, so (3 + x) / 3 = 1.Step 4: Multiply both sides by 3: 3 + x = 3, so x = 0.Step 5: Apply the centroid formula to the y coordinates: (y_A + y_B + y_C) / 3 = -4.Step 6: Substitute values: (-4 + 5 + y) / 3 = -4, which simplifies to (1 + y) / 3 = -4.Step 7: Multiply both sides by 3: 1 + y = -12, so y = -13.Step 8: Therefore, vertex C is at (0, -13).

Verification / Alternative check:
Now verify by recomputing the centroid from the three vertices A(3, -4), B(0, 5) and C(0, -13). Sum of x coordinates is 3 + 0 + 0 = 3, divided by 3 gives x coordinate 1. Sum of y coordinates is -4 + 5 - 13 = -12, divided by 3 gives -4. This matches the given centroid, so the coordinates are correct.

Why Other Options Are Wrong:
Points such as (0, 13), (0, 5), (0, -5) and (2, -13) do not satisfy both of the centroid equations simultaneously. Plugging any of these into the centroid formula fails to give (1, -4), so they cannot represent vertex C in this triangle.

Common Pitfalls:
Typical mistakes include forgetting to divide by 3 when using the centroid formula or mixing up which coordinates belong to which vertex. Some learners only match the x coordinate and forget to check the y coordinate. Working systematically with both coordinates ensures the correct solution.

Final Answer:
The coordinates of vertex C are (0, -13).