Find the coordinates of the centroid of a triangle whose vertices in the coordinate plane are A(1, −5), B(4, 0) and C(−2, 2), using the standard centroid formula.

Introduction / Context:
This question focuses on the centroid of a triangle in coordinate geometry. The centroid is the intersection point of the medians and represents the average of the vertex coordinates. It is a basic but important concept in analytic geometry.

Given Data / Assumptions:

• Vertex A = (1, −5)
• Vertex B = (4, 0)
• Vertex C = (−2, 2)
• We use the centroid formula for a triangle

Concept / Approach:
For a triangle with vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃), the centroid G has coordinates G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3). We simply sum the x coordinates, sum the y coordinates and divide each sum by three.

Step-by-Step Solution:
x coordinates: x₁ = 1, x₂ = 4, x₃ = −2 Sum of x coordinates = 1 + 4 + (−2) = 3 Centroid x coordinate = 3/3 = 1 y coordinates: y₁ = −5, y₂ = 0, y₃ = 2 Sum of y coordinates = −5 + 0 + 2 = −3 Centroid y coordinate = −3/3 = −1 So centroid G = (1, −1)

Verification / Alternative check:
We can verify by noting that the centroid must lie inside the triangle and should have coordinates between the extreme x and y values of the vertices. Here x ranges from −2 to 4 and y ranges from −5 to 2. The point (1, −1) fits comfortably within those ranges, confirming the plausibility of the result.

Why Other Options Are Wrong:
(−1, 1) simply flips the signs and does not use the centroid formula. (2, −2) and (−2, 2) are not equal to the averages of the given coordinates. (0, 0) is frequently guessed but does not reflect the asymmetric distribution of the vertices and fails the average calculations.

Common Pitfalls:
Errors usually happen when adding or dividing the coordinates, especially with negative values. Some learners incorrectly divide by 2 instead of 3, confusing the centroid with the midpoint between two points. Always remember that a triangle has three vertices so the coordinate averages must be divided by three.

Final Answer:
Therefore, the centroid of triangle ABC is (1, −1).