The coordinates of the vertex of ∆ABC are A ($a, b$), B (5, −3) and C (−5, 3). The coordinates of the centroid G of ∆ABC is $G\left(\frac{2}{3}…

(a) Given,

Vertices of ∆ABC are A ($a, b$), B (5, −3) and C (−5, 3).

Centroid G = $\Big(\frac{2}{3}, \frac{1}{3}\Big)$

By formula,

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

Here,

$x_1 = a, y_1 = b$

$x_2 = 5, y_2 = -3$

$x_3 = -5, y_3 = 3$

Substituting values we get :

$$\begin{aligned}\Rightarrow \Big(\frac{2}{3}, \frac{1}{3}\Big) = \Big(\frac{a + 5 + (-5)}{3}, \frac{b + (-3) + 3}{3}\Big) \\[1em] \Rightarrow \Big(\frac{2}{3}, \frac{1}{3}\Big) = \Big(\frac{a}{3}, \frac{b}{3}\Big).\end{aligned}$$

Comparing the coordinates,

$$\begin{aligned}\Rightarrow \frac{a}{3} = \frac{2}{3} \\[1em] \Rightarrow a = 2 \\[1em] \Rightarrow \frac{b}{3} = \frac{1}{3} \\[1em] \Rightarrow b = 1.\end{aligned}$$

Hence, the coordinates of vertex A are (2, 1).

(b) Let CM be the median, passing through C, here M is the mid-point of AB.

A = (2, 1) and B = (5, −3)

By formula,

Mid-point of AB = $\Big(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\Big)$

Here,

$x_1 = 2, y_1 = 1$

$x_2 = 5, y_2 = -3$

Substituting values we get :

$$\begin{aligned}\text{ M }= \Big(\frac{2 + 5}{2}, \frac{1 + (-3)}{2}\Big) \\[1em] = \Big(\frac{7}{2}, \frac{-2}{2}\Big) \\[1em] = \Big(\frac{7}{2}, -1\Big).\end{aligned}$$

Hence, M = $\Big(\frac{7}{2}, -1\Big)$.

The median through C is the line joining C (−5, 3) and M $\Big(\frac{7}{2}, -1\Big)$.

By formula,

Slope of a line = $\frac{y_2 - y_1}{x_2 - x_1}$

Here,

$x_1 = -5, y_1 = 3$

$x_2 = \frac{7}{2}, y_2 = -1$

Substituting values we get :

$$\begin{aligned}\text{Slope (m)} = \frac{-1 - 3}{\frac{7}{2} - (-5)} \\[1em] = \frac{-4}{\frac{7 + 10}{2}} \\[1em] = \frac{-4}{\frac{17}{2}} \\[1em] = \frac{-8}{17}.\end{aligned}$$

By point-slope form,

Equation of a line : $y - y_1 = m(x - x_1)$

Substituting values we get :

$$\begin{aligned}\Rightarrow y - 3 = \frac{-8}{17}(x - (-5)) \\[1em] \Rightarrow 17(y - 3) = -8(x + 5) \\[1em] \Rightarrow 17y - 51 = -8x - 40 \\[1em] \Rightarrow 8x + 17y - 51 + 40 = 0 \\[1em] \Rightarrow 8x + 17y - 11 = 0.\end{aligned}$$

Hence, the equation of the median through vertex C is $\bm{8x + 17y - 11 = 0}$.