Question

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

Explanation:

Step1: Find coordinates of points

Assume \(A(-6,7)\), \(B(-6,6)\), \(C(-6,6)\) (from grid).

Step2: Calculate weighted average formula

Weighted average formula for \(x\) - coordinate: \(\frac{w_1x_1 + w_2x_2+w_3x_3}{w_1 + w_2+w_3}\), for \(y\) - coordinate: \(\frac{w_1y_1 + w_2y_2+w_3y_3}{w_1 + w_2+w_3}\). Here \(w_1 = 1\), \(w_2=4\), \(w_3 = 4\), \(x_1=x_2=x_3=-6\), \(y_1 = 7\), \(y_2=y_3 = 6\).

Step3: Calculate \(x\) - coordinate of weighted average

\(\frac{1\times(-6)+4\times(-6)+4\times(-6)}{1 + 4+4}=\frac{-6-24 - 24}{9}=\frac{-54}{9}=-6\)

Step4: Calculate \(y\) - coordinate of weighted average

\(\frac{1\times7+4\times6+4\times6}{1 + 4+4}=\frac{7 + 24+24}{9}=\frac{55}{9}\approx6.11\)

Answer:

The point is \((-6,\frac{55}{9})\)