In a triangle with vertices P(x, y), Q(5, −1), and R(2, 6), the Centroid is at G(4, 2). What are the coordinates of vertex P?

Correct — A.
Centroid = average of the three vertices: G = ((x + 5 + 2) ⁄ 3, (y − 1 + 6) ⁄ 3)
x-coordinate: (x + 7) ⁄ 3 = 4, so x + 7 = 12 and x = 5
y-coordinate: (y + 5) ⁄ 3 = 2, so y + 5 = 6 and y = 1

P = (5, 1) → option (a)

Why the others are wrong

• (b)
  (5, −1) — (5, −1) repeats Q's coordinates, so P and Q would be one point. The x-average still gives 4, but the y-average is (−1 − 1 + 6) ⁄ 3 = 4⁄3, not 2.
• (c)
  (4, 1) — (4, 1) gets y right but not x: (4 + 5 + 2) ⁄ 3 = 11⁄3, not 4.
• (d)
  (4, 2) — (4, 2) is the centroid G itself, not a vertex. As P it gives x-average 11⁄3 and y-average (2 − 1 + 6) ⁄ 3 = 7⁄3, neither matching G(4, 2).

The centroid is where the three medians meet, and its coordinates are the averages of the vertices: G = ((x₁ + x₂ + x₃) ⁄ 3, (y₁ + y₂ + y₃) ⁄ 3).

Run it backwards to find a missing vertex: P = 3G − Q − R. Here 3G = (12, 6), so P = (12 − 5 − 2, 6 − (−1) − 6) = (5, 1).

Key facts

• Centroid of (x₁, y₁), (x₂, y₂), (x₃, y₃) = ((x₁ + x₂ + x₃) ⁄ 3, (y₁ + y₂ + y₃) ⁄ 3)
• Missing vertex = 3 × centroid − the sum of the two known vertices
• The centroid divides each median in the ratio 2 : 1 from the vertex

Common traps

• Carrying Q's y-value as +1 instead of −1: (y + 1 + 6) ⁄ 3 = 2 then gives y = −1 and the point (5, −1).
• Answering with the centroid's coordinates when the question asks for a vertex.

No directly related past PYQ was found.