If A(5,−1),B(−3,−2) and C(−1,8) are the vertices of triangle ABC, find the length of median through A and the coordinates of the centroid.

\$A(5,−1), B(−3,−2)\$ and \$C(−1,8)\$ are the vertices of triangle ABC.

Let \$(x_1 , y_1) =(5,-1) \$, \$(x_2 , y_2) =(-3, -2 ) \$ and \$(x_3, y_3) =(-1, 8 ) \$

The centroid of the triangle is given by formula

\begin{align} \require{cancel} && Centroid = \Bigl(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \Bigr) \end{align}

coincide \begin{align} \require{cancel} \therefore && \Bigl( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \Bigr) &= \Biggl( \frac{5 - 3 - 1}{3}, \frac{-1 - 2 -1}{3} \Biggr) \\ \Rightarrow && &= \Biggl( \frac{1}{3}, \frac{5}{3} \Biggr) \\ \end{align}

Distance between A and centroid is

\$\frac{2 \sqrt{65} }{3}\$

Since centroid divides median in the ratio 2:1

Length of median

\begin{align} \require{cancel} &= \frac{3}{2} \times \frac{2 \sqrt{65} }{3} \\ &= \sqrt{65} \end{align}

Answer

The equation of a straight line perpendicular to the line joining the points (4,5) and (1,2) and passing through the point (3,2) is \$ x + y - 5 = 0\$