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

Answer

First, identify the coordinates of the points:
- From the graph, point (A=(7,7)), point (B=(6, - 4)), and point (C=(-3,-3)).
- The formula for the weighted - average of points ((x_1,y_1),(x_2,y_2),(x_3,y_3)) with weights (w_1,w_2,w_3) is ((\frac{w_1x_1 + w_2x_2+w_3x_3}{w_1 + w_2+w_3},\frac{w_1y_1 + w_2y_2+w_3y_3}{w_1 + w_2+w_3})).
- Here, (w_1 = 1), (w_2 = 4), (w_3 = 3), (x_1 = 7), (x_2 = 6), (x_3=-3), (y_1 = 7), (y_2=-4), (y_3=-3).
Calculate the (x) - coordinate of the weighted average:
- Step1: Calculate the numerator of the (x) - coordinate formula
  - (w_1x_1+w_2x_2 + w_3x_3=1\times7 + 4\times6+3\times(-3))
  - (=7 + 24-9)
  - (=22).
- Step2: Calculate the denominator of the (x) - coordinate formula
  - (w_1 + w_2+w_3=1 + 4+3=8).
- Step3: Calculate the (x) - coordinate of the weighted average
  - The (x) - coordinate of the weighted average is (\frac{w_1x_1 + w_2x_2+w_3x_3}{w_1 + w_2+w_3}=\frac{22}{8}=\frac{11}{4}=2.75).
Calculate the (y) - coordinate of the weighted average:
- Step1: Calculate the numerator of the (y) - coordinate formula
  - (w_1y_1+w_2y_2 + w_3y_3=1\times7 + 4\times(-4)+3\times(-3))
  - (=7-16 - 9)
  - (=7-(16 + 9)=7 - 25=-18).
- Step2: Calculate the denominator of the (y) - coordinate formula
  - (w_1 + w_2+w_3 = 8).
- Step3: Calculate the (y) - coordinate of the weighted average
  - The (y) - coordinate of the weighted average is (\frac{w_1y_1 + w_2y_2+w_3y_3}{w_1 + w_2+w_3}=\frac{-18}{8}=-\frac{9}{4}=-2.25).

((2.75,-2.25))