Chapter 9.5, Problem 39E

a.

To determine

Toshow:Civic center is at midpoint of hospital and stadium.

Explanation of Solution

Given information:

Location of hospital- $\left(x_1,y_1\right)=\left(1,2\right)$

Location of stadium- $\left(x_2,y_2\right)=\left(9,6\right)$

Location of civic center- $\left(x_3,y_3\right)=\left(5,4\right)$

Concept Used:

Midpoint of two points - $M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Calculation:

Midpoint of hospital and stadium will be given by-

  $\begin{array}{l}M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ \Rightarrow M=\left(\frac{1+9}{2},\frac{2+6}{2}\right)\\ \Rightarrow M=\left(\frac{10}{2},\frac{8}{2}\right)\\ \Rightarrow M=\left(5,4\right)\end{array}$

  $\because M=\left(5,4\right)=\left(x_3,y_3\right)$

Therefore, we can say that civic center is halfway between hospital and stadium.

b.

To determine

Tocompare:Distance of civic center and stadium with hospital and state house.

Explanation of Solution

Given information:

Location of hospital- $\left(x_1,y_1\right)=\left(1,2\right)$

Location of stadium- $\left(x_2,y_2\right)=\left(9,6\right)$

Location of civic center- $\left(x_3,y_3\right)=\left(5,4\right)$

Location of state house- $\left(x_4,y_4\right)=\left(1,7\right)$

Concept Used:

Distance between two points - $d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Calculation:

Distance of hospital and statehouse will be given by-

  $\begin{array}{l}{d}_1=\sqrt{{\left(x_4-x_1\right)}^2+{\left(y_4-y_1\right)}^2}\\ \Rightarrow d_1=\sqrt{{\left(1-1\right)}^2+{\left(7-2\right)}^2}\\ \Rightarrow d_1=\sqrt{{\left(0\right)}^2+{\left(5\right)}^2}\\ \Rightarrow d_1=\sqrt{0+25}\\ \Rightarrow d_1=\sqrt{25}\\ \Rightarrow d_1=5\end{array}$

Distance of civic center and stadium will be given by-

  $\begin{array}{l}{d}_2=\sqrt{{\left(x_2-x_3\right)}^2+{\left(y_2-y_3\right)}^2}\\ \Rightarrow d_2=\sqrt{{\left(9-5\right)}^2+{\left(6-4\right)}^2}\\ \Rightarrow d_2=\sqrt{{\left(4\right)}^2+{\left(2\right)}^2}\\ \Rightarrow d_2=\sqrt{16+4}\\ \Rightarrow d_2=\sqrt{20}\end{array}$

Therefore, we can say that hospital and state house compared to civic center and stadium.