The coordinates of the centroid of triangle ABC are (−1, −2). If the coordinates of vertices A and B are (6, −4) and (−2, 2) respectively, what are the coordinates of vertex C?

Introduction / Context:
This coordinate geometry problem involves the centroid of a triangle. The centroid is the point where the three medians intersect and is known to be the average of the coordinates of the three vertices. We are given the centroid and two vertices and must find the coordinates of the third vertex.

Given Data / Assumptions:
Centroid G has coordinates (−1, −2).Vertex A has coordinates (6, −4).Vertex B has coordinates (−2, 2).Vertex C has coordinates (x, y), which we must determine.

Concept / Approach:
The centroid of a triangle with vertices A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) is given by G = ((x₁ + x₂ + x₃) / 3, (y₁ + y₂ + y₃) / 3). We know G, A and B, so we can set up equations for the x and y coordinates of G and solve for x and y, the coordinates of C.

Step-by-Step Solution:
Step 1: Let C be (x, y).Step 2: Use the centroid formula for the x coordinate: (xA + xB + xC) / 3 = xG.Step 3: Substitute values: (6 + (−2) + x) / 3 = −1.Step 4: Simplify the numerator: (4 + x) / 3 = −1, so 4 + x = −3.Step 5: Solve for x: x = −3 − 4 = −7.Step 6: Now use the centroid formula for the y coordinate: (yA + yB + yC) / 3 = yG.Step 7: Substitute values: (−4 + 2 + y) / 3 = −2.Step 8: Simplify the numerator: (−2 + y) / 3 = −2, so −2 + y = −6.Step 9: Solve for y: y = −6 + 2 = −4.Step 10: Therefore, C has coordinates (−7, −4).

Verification / Alternative check:
Check the centroid using A(6, −4), B(−2, 2) and C(−7, −4). The x coordinate of the centroid is (6 + (−2) + (−7)) / 3 = (−3) / 3 = −1. The y coordinate is (−4 + 2 + (−4)) / 3 = (−6) / 3 = −2. This matches the given centroid (−1, −2), confirming the correctness of C(−7, −4).

Why Other Options Are Wrong:
The other options change signs or swap coordinates, giving centroids different from (−1, −2) when plugged into the formula. For example, using (7, −4) or (7, 4) would yield a positive x centroid, which contradicts the given centroid. Only (−7, −4) produces the correct average.

Common Pitfalls:
Some learners incorrectly divide by 2 instead of 3 or forget to include all three vertices when averaging. Others may make sign errors when adding the coordinates. Carefully applying the centroid formula and checking both coordinates prevents these mistakes.

Final Answer:
The coordinates of vertex C are (−7, −4).