The centroid of triangle ABC has coordinates (3, −2). If vertices A and B are at A(−2, 5) and B(6, −2) respectively, determine the exact coordinates of vertex C.

Introduction / Context:
This coordinate geometry question uses the formula for the centroid of a triangle. The centroid is the point where the three medians intersect and is located at the average of the x coordinates and the average of the y coordinates of the three vertices. Given the centroid and two vertex coordinates, you can work backwards to find the third vertex. This idea is useful in analytic geometry and appears frequently in exam problems.

Given Data / Assumptions:

• Triangle ABC has centroid G with coordinates (3, −2).
• Vertex A is at (−2, 5).
• Vertex B is at (6, −2).
• Vertex C has coordinates (x, y) which we need to determine.
• All coordinates are real numbers.

Concept / Approach:
The centroid G of triangle ABC with vertices A(x1, y1), B(x2, y2), and C(x3, y3) is given by G = ( (x1 + x2 + x3)/3 , (y1 + y2 + y3)/3 ). Here, G is known, and two vertices are known, so x3 and y3 can be solved by reversing the centroid formulas. We set up two equations, one for the x coordinates and one for the y coordinates, and solve them for the unknown coordinates of vertex C.

Step-by-Step Solution:
Step 1: Let C have coordinates (x, y).Step 2: Apply the centroid formula for the x coordinate: (xA + xB + xC)/3 = 3.Step 3: Substitute xA = −2 and xB = 6 to get (−2 + 6 + x)/3 = 3.Step 4: Simplify the numerator: (4 + x)/3 = 3, so 4 + x = 9 and x = 5.Step 5: Now use the centroid formula for the y coordinate: (yA + yB + yC)/3 = −2.Step 6: Substitute yA = 5 and yB = −2 to obtain (5 − 2 + y)/3 = −2.Step 7: Simplify: (3 + y)/3 = −2, so 3 + y = −6 and y = −9.Step 8: Thus C has coordinates (5, −9).

Verification / Alternative check:
Check by recomputing the centroid using A(−2, 5), B(6, −2), and C(5, −9). The average of the x coordinates is (−2 + 6 + 5)/3 = 9/3 = 3. The average of the y coordinates is (5 − 2 − 9)/3 = (−6)/3 = −2. These match the given centroid coordinates, confirming that C(5, −9) is correct.

Why Other Options Are Wrong:
Points such as (−5, −9), (5, 9), and (−5, 9) produce different averages for x or y when used in the centroid formula, so they cannot give G(3, −2). The point (3, −2) is the centroid itself rather than a vertex and therefore cannot be vertex C of triangle ABC. Only the coordinate (5, −9) reproduces the correct centroid when combined with A and B.

Common Pitfalls:
Some students confuse the centroid formula with the midpoint formula, averaging only two points instead of three. Others incorrectly multiply or divide by 3 when rearranging equations. Writing the centroid equations carefully for x and y separately and solving step by step helps avoid arithmetic mistakes and ensures the correct coordinates for the missing vertex.

Final Answer:
The coordinates of vertex C are (5, −9).