what is the average of the points a, b and c with weights 3, 1 and 2 respectively?

Explanation:

Step1: Identify coordinates of points

$A=(7,2)$, $B=(0,6)$, $C=(7, - 8)$

Step2: Calculate weighted - average of x - coordinates

The formula for the weighted average of $x$ - coordinates $\bar{x}=\frac{w_1x_1 + w_2x_2+w_3x_3}{w_1 + w_2+w_3}$, where $w_1 = 3$, $x_1 = 7$, $w_2 = 1$, $x_2 = 0$, $w_3 = 2$, $x_3 = 7$. $\bar{x}=\frac{3\times7+1\times0 + 2\times7}{3 + 1+2}=\frac{21+0 + 14}{6}=\frac{35}{6}\approx5.83$

Step3: Calculate weighted - average of y - coordinates

The formula for the weighted average of $y$ - coordinates $\bar{y}=\frac{w_1y_1 + w_2y_2+w_3y_3}{w_1 + w_2+w_3}$, where $w_1 = 3$, $y_1 = 2$, $w_2 = 1$, $y_2 = 6$, $w_3 = 2$, $y_3=-8$. $\bar{y}=\frac{3\times2+1\times6+2\times(-8)}{3 + 1+2}=\frac{6 + 6-16}{6}=\frac{-4}{6}=-\frac{2}{3}\approx - 0.67$

Answer:

$(\frac{35}{6},-\frac{2}{3})$