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?
(2, 3)
The centroid of a triangle with vertices \(P(x_1,y_1),\ Q(x_2,y_2),\ R(x_3,y_3)\) is
\(G = \left(\dfrac{x_1 + x_2 + x_3}{3},\ \dfrac{y_1 + y_2 + y_3}{3}\right)\).
x-coordinate of R:
\(1 = \dfrac{3 + (-2) + x_3}{3} \;\Rightarrow\; 3 = 1 + x_3 \;\Rightarrow\; x_3 = 2\).
y-coordinate of R:
\(2 = \dfrac{-1 + 4 + y_3}{3} \;\Rightarrow\; 6 = 3 + y_3 \;\Rightarrow\; y_3 = 3\).
Hence \(R = (2,\ 3)\).
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