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

Explanation:

Step1: Identify coordinates

Assume (A=(0,4)), (B=(2, - 6)), (C=(5,6)). We will calculate the weighted - average of the (x) - coordinates and (y) - coordinates separately.

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 = 1), (x_1 = 0), (w_2 = 4), (x_2 = 2), (w_3 = 4), (x_3 = 5). (\bar{x}=\frac{1\times0 + 4\times2+4\times5}{1 + 4+4}=\frac{0 + 8 + 20}{9}=\frac{28}{9})

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 = 1), (y_1 = 4), (w_2 = 4), (y_2=-6), (w_3 = 4), (y_3 = 6). (\bar{y}=\frac{1\times4+4\times(-6)+4\times6}{1 + 4+4}=\frac{4-24 + 24}{9}=\frac{4}{9})

Answer:

((\frac{28}{9},\frac{4}{9}))