Question

what is the average of the points a, b and c with weights 3, 1 and 4 respectively?

Explanation:

Step1: Identify coordinates

$A(-7,-3)$, $B(-5,3)$, $C(7,2)$

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 $w_1 = 3$, $x_1=-7$, $w_2 = 1$, $x_2=-5$, $w_3 = 4$, $x_3 = 7$.
$\bar{x}=\frac{3\times(-7)+1\times(-5)+4\times7}{3 + 1+4}=\frac{-21-5 + 28}{8}=\frac{2}{8}=\frac{1}{4}$

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 $w_1 = 3$, $y_1=-3$, $w_2 = 1$, $y_2=3$, $w_3 = 4$, $y_3 = 2$.
$\bar{y}=\frac{3\times(-3)+1\times3+4\times2}{3 + 1+4}=\frac{-9 + 3+8}{8}=\frac{2}{8}=\frac{1}{4}$

Answer:

$(\frac{1}{4},\frac{1}{4})$