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

Explanation:

Step1: Determine the coordinates of points

Let's assume the coordinates of point $A$ are $(-3,-3)$, point $B$ are $(-3,-6)$ and point $C$ are $(-7,2)$.

Step2: Calculate the 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 $w_1 = 1$, $x_1=-3$, $w_2 = 1$, $x_2=-3$, $w_3 = 3$, $x_3=-7$. $\bar{x}=\frac{1\times(-3)+1\times(-3)+3\times(-7)}{1 + 1+3}=\frac{-3-3 - 21}{5}=\frac{-27}{5}=-5.4$

Step3: Calculate the 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 $w_1 = 1$, $y_1=-3$, $w_2 = 1$, $y_2=-6$, $w_3 = 3$, $y_3=2$. $\bar{y}=\frac{1\times(-3)+1\times(-6)+3\times2}{1 + 1+3}=\frac{-3-6 + 6}{5}=\frac{-3}{5}=-0.6$

Answer:

$(-5.4,-0.6)$