Question

build perseverance find the location of the weighted average of $overline{ad}$ such that point $a$ weighs twice as much as point $c$, point $b$ weighs three times as much as point $c$, and point $d$ weighs twice as much as point $a$. if point $c$ has a weight of 2, what are the coordinates of the weighted average?

Explanation:

Step1: Determine the weights of each point

Given $w_C = 2$. Since point $A$ weighs twice as much as point $C$, then $w_A=2\times w_C = 4$. Since point $B$ weighs three times as much as point $C$, then $w_B = 3\times w_C=6$. Since point $D$ weighs twice as much as point $A$, then $w_D=2\times w_A = 8$.

Step2: Find the coordinates of each point

From the graph, $A=(-5,8)$, $B=(-1,2)$, $C=(1, - 7)$, $D=(3,-10)$.

Step3: Calculate the weighted - average of the x - coordinates

The formula for the weighted - average of the x - coordinates $\bar{x}=\frac{w_Ax_A + w_Bx_B+w_Cx_C + w_Dx_D}{w_A + w_B+w_C + w_D}$.
Substitute the values:
\[

$$\begin{align*} \bar{x}&=\frac{4\times(-5)+6\times(-1)+2\times1 + 8\times3}{4 + 6+2 + 8}\\ &=\frac{-20-6 + 2+24}{20}\\ &=\frac{-26 + 26}{20}\\ &=0 \end{align*}$$

\]

Step4: Calculate the weighted - average of the y - coordinates

The formula for the weighted - average of the y - coordinates $\bar{y}=\frac{w_Ay_A + w_By_B+w_Cy_C + w_Dy_D}{w_A + w_B+w_C + w_D}$.
Substitute the values:
\[

$$\begin{align*} \bar{y}&=\frac{4\times8+6\times2+2\times(-7)+8\times(-10)}{4 + 6+2 + 8}\\ &=\frac{32 + 12-14-80}{20}\\ &=\frac{44-94}{20}\\ &=\frac{-50}{20}\\ &=-\frac{5}{2}=-2.5 \end{align*}$$

\]

Answer:

$(0,-2.5)$