If a triangle has its centroid at \$(4,3)\$ and two of its vertices are \$(2,–1)\$ and \$(7,8)\$,find the third vertex.

Here \$(x_1, y_1) = (2, -1), (x_2, y_2) = (7, 8)\$.

Let the third vertex be (a,b) (\$(x_3, y_3) = (a, b)\$) then the centroid of the triangle is

The centroid of the triangle is given by formula

\$ ( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} ) \$

Substiutute the values.

\$\Rightarrow ( \frac{2 + 7 + a}{3}, \frac{(-1) + 8 + b}{3} ) \$

Add.

\$ \Rightarrow (\frac{9 + a}{3}, \frac{7 + b}{3} ) \$

Equating coordiantes.

\begin{align} \require{cancel} \frac{9 + a}{3} &= 4, & \frac{ 7 + b}{3} 7 &= 3 \\ 9 + a &= 12, & 7 + b &= 9 \\ a &= 12 - 9, & b &= 9 - 7 \\ a &= 3, & b &= 2 \\ \end{align}

Answer

The centroid of the triangle is \$(4, 2)\$.