Two vertices of a triangle are (3, – 5) and (- 7,4). If its centroid is (2, -1), then the third vertex is _______________.

We know, the centroid of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) = $\left(\frac{{\mathrm{x}}_1 + {\mathrm{x}}_2 + {\mathrm{x}}_3}{3} , \frac{{\mathrm{y}}_1 + {\mathrm{y}}_2 + {\mathrm{y}}_3}{3} \right)$

Let us consider the coordinates of the third vertex to be (x, y).

 $\text{⇒ (}\frac{\mathrm{x}+3-7}{3},\frac{\mathrm{y}-5+4}{3})=(2,-1)$

$\text{⇒ (}\frac{\mathrm{x}-4}{3},\frac{\mathrm{y}-1}{3})=(2,-1)$

By comparing both sides, we get:

 $\Rightarrow \frac{\mathrm{x}-4}{3}=2 , \frac{\mathrm{y}-1}{3}=-1$

$\Rightarrow x - 4 = 6, y - 1 = -3$

$\Rightarrow x = 6 + 4, y = -3 + 1$

$\Rightarrow x = 10, y = -2$

Thus, the third vertex = (10, - 2)