Question

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

Explanation:

Step1: Identify the coordinates

Let \(A=(3, 3)\), \(B = (- 5,-5)\), \(C=(-7,3)\).

Step2: Use the weighted - average formula 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 \(w_1 = 1\), \(x_1 = 3\), \(w_2 = 4\), \(x_2=-5\), \(w_3 = 1\), \(x_3=-7\).
\(\bar{x}=\frac{1\times3+4\times(-5)+1\times(-7)}{1 + 4+1}=\frac{3-20 - 7}{6}=\frac{3-(20 + 7)}{6}=\frac{3 - 27}{6}=\frac{-24}{6}=-4\).

Step3: Use the weighted - average formula 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 \(w_1 = 1\), \(y_1 = 3\), \(w_2 = 4\), \(y_2=-5\), \(w_3 = 1\), \(y_3=3\).
\(\bar{y}=\frac{1\times3+4\times(-5)+1\times3}{1 + 4+1}=\frac{3-20 + 3}{6}=\frac{(3 + 3)-20}{6}=\frac{6-20}{6}=\frac{-14}{6}=-\frac{7}{3}\).

Answer:

\((-4,-\frac{7}{3})\)