Question

harrison’s house is halfway between the library and town hall. if harrison’s house is at (6, -2) on a coordinate plane and the library is at (4, -9), what is the location of town hall? enter the correct coordinates in the boxes.

Explanation:

Step1: Recall midpoint formula

The midpoint formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(M = (\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\), where \(M\) is the midpoint. Let the coordinates of the town hall be \((x,y)\), the library be \((x_1,y_1)=(4, - 9)\) and the midpoint (Harrison's house) be \((x_m,y_m)=(6,-2)\).

Step2: Solve for \(x\) (x - coordinate of town hall)

Using the midpoint formula for \(x\) - coordinate: \(\frac{x_1+x}{2}=x_m\). Substitute \(x_1 = 4\) and \(x_m=6\) into the formula:
\[

$$\begin{align*} \frac{4 + x}{2}&=6\\ 4+x&=12\\ x&=12 - 4\\ x&=8 \end{align*}$$

\]

Step3: Solve for \(y\) (y - coordinate of town hall)

Using the midpoint formula for \(y\) - coordinate: \(\frac{y_1 + y}{2}=y_m\). Substitute \(y_1=-9\) and \(y_m = - 2\) into the formula:
\[

$$\begin{align*} \frac{-9 + y}{2}&=-2\\ -9 + y&=-4\\ y&=-4 + 9\\ y&=5 \end{align*}$$

\]

Answer:

\((8,5)\)