The coordinates of the centroid of triangle ABC are (2, 2). If the coordinates of vertices A and B are (7, −1) and (1, 2) respectively, what are the coordinates of vertex C?

Introduction / Context:
This question uses the centroid formula from coordinate geometry. The centroid of a triangle is the average of the coordinates of its three vertices. If the centroid and two vertices are known, you can reverse this relationship to find the third vertex. This is a useful technique in both pure math problems and practical geometry applications.

Given Data / Assumptions:

• Triangle ABC has centroid G with coordinates (2, 2).
• Vertex A has coordinates (7, −1).
• Vertex B has coordinates (1, 2).
• Vertex C has unknown coordinates (x, y).
• The centroid of a triangle with vertices (x1, y1), (x2, y2) and (x3, y3) is ((x1 + x2 + x3) / 3, (y1 + y2 + y3) / 3).

Concept / Approach:
We use the centroid formula in reverse. Since we know the final centroid coordinates and two of the three vertices, we set up two equations: one for the x coordinate and one for the y coordinate. Solving these linear equations gives the missing coordinates of vertex C. This is a direct algebraic manipulation of the centroid definition.

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 = 2. Step 3: Substitute xA = 7, xB = 1 and xC = x: (7 + 1 + x) / 3 = 2. Step 4: Simplify: (8 + x) / 3 = 2 ⇒ 8 + x = 6 ⇒ x = 6 − 8 = −2. Step 5: Apply the centroid formula for the y coordinate: (yA + yB + yC) / 3 = 2. Step 6: Substitute yA = −1, yB = 2 and yC = y: (−1 + 2 + y) / 3 = 2. Step 7: Simplify: (1 + y) / 3 = 2 ⇒ 1 + y = 6 ⇒ y = 6 − 1 = 5. Step 8: Therefore, vertex C has coordinates (−2, 5).

Verification / Alternative check:
Check the centroid calculation explicitly. Average of x coordinates: (7 + 1 + (−2)) / 3 = (6) / 3 = 2. Average of y coordinates: (−1 + 2 + 5) / 3 = 6 / 3 = 2. Both match the given centroid (2, 2), confirming that our computed point C(−2, 5) is correct.

Why Other Options Are Wrong:
(2, 5), (−2, −5), (2, −5) and (0, 5) all fail when substituted into the centroid formula; the resulting average coordinates are not (2, 2). For each incorrect option, either the x average or the y average (or both) deviate from 2, so they cannot represent the true position of vertex C consistent with the given centroid.

Common Pitfalls:
A typical mistake is to mix up the formula, for example dividing by 2 instead of 3, or averaging only two vertices. Another common issue is arithmetic error when solving the simple linear equations, especially with negative numbers. Writing down the equations carefully and checking each substitution helps avoid such slips.

Final Answer:
The coordinates of vertex C are (−2, 5).