In a triangle PQR, the centroid is at G(1, 2). If the vertices P and Q are at (3, -1) and (-2, 4) respectively, what are the coordinates of vertex R?

Correct — D.
Centroid rule: G = ((x₁ + x₂ + x₃)⁄3, (y₁ + y₂ + y₃)⁄3)
Rearrange: x₃ = 3 × (x of G) − x₁ − x₂, and the same for y

x of R: 3 × 1 − 3 − (−2) = 3 − 3 + 2 = 2
y of R: 3 × 2 − (−1) − 4 = 6 + 1 − 4 = 3

Check: (3 − 2 + 2)⁄3 = 1 and (−1 + 4 + 3)⁄3 = 2, which is G(1, 2)
So R = (2, 3) → option (d)

Why the others are wrong

• (a)
  (2, 5) — The y-values must total 3 × 2 = 6. P and Q already give −1 + 4 = 3, so R's y is 3, not 5. The x-value 2 is right.
• (b)
  (2, 4) — y = 4 overshoots by 1. With R = (2, 4) the centroid's y would be (−1 + 4 + 4)⁄3 = 7⁄3, not 2. The x-value 2 is right.
• (c)
  (1, 3) — x = 1 fails. The x-values must total 3 × 1 = 3, and P and Q give 3 + (−2) = 1, so R's x is 2. The y-value 3 is right.

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

To find a missing vertex, undo the average: multiply the centroid's coordinate by 3, then subtract the two known vertices. Work x and y separately.

The centroid also divides each median in the ratio 2 : 1 from the vertex, which is how an item giving one vertex and asking for the midpoint of the opposite side is solved.

Key facts

• Centroid of (x₁, y₁), (x₂, y₂), (x₃, y₃) = ((x₁ + x₂ + x₃)⁄3, (y₁ + y₂ + y₃)⁄3)
• Missing vertex: x₃ = 3 × (x of G) − x₁ − x₂, and likewise for y
• The centroid divides each median in the ratio 2 : 1, measured from the vertex

Common traps

• Dropping the sign when subtracting a negative coordinate: 3 − 3 − (−2) is 2, not −2
• Forgetting to multiply G by 3: 1 − 3 − (−2) = 0 gives a wrong x-value

No directly related past PYQ was found.