Question

what are the coordinates of a point p that represents the weighted average of point a and point b if point b weighs three times as much as point a?

Explanation:

Step1: Find coordinates of A and B

From the graph, \(A(-2,4)\) and \(B(4,1)\). Let the weight of \(A = w_{A}=1\) and weight of \(B = w_{B} = 3\).

Step2: Use weighted - average formula for x - coordinate

The formula for the x - coordinate of the weighted average \(x=\frac{w_{A}x_{A}+w_{B}x_{B}}{w_{A}+w_{B}}\). Substitute \(x_{A}=-2\), \(x_{B}=4\), \(w_{A}=1\), \(w_{B}=3\):
\(x=\frac{1\times(-2)+3\times4}{1 + 3}=\frac{-2 + 12}{4}=\frac{10}{4}=\frac{5}{2}=2.5\)

Step3: Use weighted - average formula for y - coordinate

The formula for the y - coordinate of the weighted average \(y=\frac{w_{A}y_{A}+w_{B}y_{B}}{w_{A}+w_{B}}\). Substitute \(y_{A}=4\), \(y_{B}=1\), \(w_{A}=1\), \(w_{B}=3\):
\(y=\frac{1\times4+3\times1}{1 + 3}=\frac{4+3}{4}=\frac{7}{4}=1.75\)

Answer:

\((2.5,1.75)\)