Question

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

Explanation:

Step1: Identify coordinates

From the graph, point $A=(0,7)$, point $B=(3, - 3)$, point $C=(7,8)$.

Step2: Calculate weighted - average for 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 $w_1 = 1$, $x_1 = 0$, $w_2 = 1$, $x_2 = 3$, $w_3 = 3$, $x_3 = 7$.
$\bar{x}=\frac{1\times0 + 1\times3+3\times7}{1 + 1+3}=\frac{0 + 3+21}{5}=\frac{24}{5}=4.8$

Step3: Calculate weighted - average for 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 $w_1 = 1$, $y_1 = 7$, $w_2 = 1$, $y_2=-3$, $w_3 = 3$, $y_3 = 8$.
$\bar{y}=\frac{1\times7+1\times(-3)+3\times8}{1 + 1+3}=\frac{7-3 + 24}{5}=\frac{28}{5}=5.6$

Answer:

$(4.8,5.6)$