The co ordinates of the centroid of triangle ABC are given as (−1, 4). If the co ordinates of vertices A and B are (−3, −1) and (3, 5) respectively, what are the co ordinates of vertex C?

Introduction / Context:
This coordinate geometry problem involves the centroid of a triangle. The centroid is the point where the three medians of a triangle intersect, and it has a simple formula in terms of the vertex co ordinates. In this question, the centroid and two vertices are known, and the task is to determine the co ordinates of the third vertex. This type of problem is very standard in analytic geometry and tests the ability to use and rearrange the centroid formula.

Given Data / Assumptions:
- Triangle ABC has centroid G with co ordinates (−1, 4). - Vertex A has co ordinates (−3, −1). - Vertex B has co ordinates (3, 5). - Vertex C has co ordinates (xC, yC), which we must find. - The centroid formula for a triangle with vertices (xA, yA), (xB, yB), (xC, yC) is used.

Concept / Approach:
The centroid of a triangle is the average of the x co ordinates and the average of the y co ordinates of its vertices. If G is the centroid, then xG = (xA + xB + xC) / 3 and yG = (yA + yB + yC) / 3. In this question, xG and yG are known along with xA, yA, xB, and yB. Therefore, we can set up two linear equations in xC and yC, solve them, and match the result with the provided options.

Step-by-Step Solution:
Step 1: Write down the centroid formulas. Step 2: For the x co ordinate: xG = (xA + xB + xC) / 3. Step 3: Substitute the known values: xG = −1, xA = −3, xB = 3. So we have −1 = (−3 + 3 + xC) / 3. Step 4: Simplify inside the bracket: −3 + 3 + xC = xC. So −1 = xC / 3. Step 5: Multiply both sides by 3 to get xC = −3. Step 6: For the y co ordinate: yG = (yA + yB + yC) / 3. Step 7: Substitute the known values: yG = 4, yA = −1, yB = 5. So 4 = (−1 + 5 + yC) / 3. Step 8: Simplify −1 + 5 + yC = 4 + yC. So 4 = (4 + yC) / 3. Step 9: Multiply both sides by 3 to get 12 = 4 + yC. Therefore yC = 12 − 4 = 8. Step 10: Hence the co ordinates of C are (−3, 8).

Verification / Alternative check:
As a quick check, substitute A(−3, −1), B(3, 5), and C(−3, 8) back into the centroid formula. Compute the average of the x co ordinates: (−3 + 3 − 3) / 3 = (−3) / 3 = −1. Compute the average of the y co ordinates: (−1 + 5 + 8) / 3 = 12 / 3 = 4. This matches the given centroid (−1, 4), so the answer is verified.

Why Other Options Are Wrong:
Option (3, 8) would give an x average different from −1 and thus does not produce the correct centroid. Option (−3, −8) yields a y average that is negative and far from 4, so it cannot be correct. Option (3, −8) fails both the x and y average checks and does not give the required centroid.

Common Pitfalls:
Students sometimes confuse the centroid with other special points such as the circumcentre or orthocentre and attempt to use incorrect formulas. Another common error is arithmetic mistakes when summing the co ordinates or dividing by 3. To avoid such errors, always write the centroid formula clearly, substitute carefully, and check by recomputing the centroid from the final vertex co ordinates.

Final Answer:
The co ordinates of vertex C are (−3, 8).