what is the average of the points a, b and c with weights 1, 4 and 2 respectively?

Explanation:

Step1: Identify coordinates

From the graph, $A=(2,5)$, $B = (- 6,-3)$, $C=(1,-4)$.

Step2: Calculate weighted - average of x - coordinates

The formula for the weighted - average of $x$ values $x_{avg}=\frac{w_1x_1 + w_2x_2+w_3x_3}{w_1 + w_2+w_3}$, where $w_1 = 1$, $x_1=2$, $w_2 = 4$, $x_2=-6$, $w_3 = 2$, $x_3 = 1$. $x_{avg}=\frac{1\times2+4\times(-6)+2\times1}{1 + 4+2}=\frac{2-24 + 2}{7}=\frac{-20}{7}\approx - 2.86$.

Step3: Calculate weighted - average of y - coordinates

The formula for the weighted - average of $y$ values $y_{avg}=\frac{w_1y_1 + w_2y_2+w_3y_3}{w_1 + w_2+w_3}$, where $w_1 = 1$, $y_1 = 5$, $w_2 = 4$, $y_2=-3$, $w_3 = 2$, $y_3=-4$. $y_{avg}=\frac{1\times5+4\times(-3)+2\times(-4)}{1 + 4+2}=\frac{5-12 - 8}{7}=\frac{-15}{7}\approx - 2.14$.

Answer:

$\left(-\frac{20}{7},-\frac{15}{7}\right)$