Find the value of x for which the triangle with vertices (2, 2), (4, 2) and (x, 8) will have the centroid (5, 5).

 

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

Given vertices of the triangle = (2, 2), (4, 2), and (x, 8) 

Given centroid = (5, 5)

Thus, 

$(\frac{2+4+\mathrm{x}}{3},\frac{2+2+8}{3})=(5, 5)$

Comparing the x coordinates, we get:

$\Rightarrow \frac{2+4+\mathrm{x}}{3}=5$

$\Rightarrow 6 + \text{x}=15$

$\Rightarrow \text{x}=15 - 6$

$\Rightarrow \text{x = 9}$