Chapter 1.1, Problem 31E

To determine

Tograph : the given points in the coordinate axis.

Explanation of Solution

Given information : we have the given points are $\left(1,1\right)and\left(9,7\right)$

Graph :

The plotting of the points is shown in the graph below

  

Interpretation :

The point $\left(1,1\right)$ is shown by blue dot and point $\left(9,7\right)$ is shown by red dot.

To determine

To find : the distance between two given points.

Answer to Problem 31E

The distance between given points is $10$ units.

Explanation of Solution

Given information : we have the given points are $\left(1,1\right)and\left(9,7\right)$

Let the points be $A\left(1,1\right)andB\left(9,7\right)$

We know that the distance between two points $A\left(x_1,y_1\right)andB\left(x_2,y_2\right)$ is given by

  $AB=\sqrt{{\left(x_1-x_2\right)}^2+{\left(y_1-y_2\right)}^2}$

Now, using the distance formula we get

  $\begin{array}{c}AB=\sqrt{{\left(1-9\right)}^2+{\left(1-7\right)}^2}\\ =\sqrt{{\left(-8\right)}^2+{\left(-6\right)}^2}\\ =\sqrt{64+36}\\ =\sqrt{100}\\ =10\end{array}$

Therefore, the distance between given points is $10$ units.

To determine

To find : the midpoint of the line segment joining the points

Answer to Problem 31E

The midpoint between given points is $P\left(x,y\right)=\left(5,4\right)$

Explanation of Solution

Given information : we have the given points are $\left(1,1\right)and\left(9,7\right)$

Let the points be $A\left(1,1\right)andB\left(9,7\right)$ and $P\left(x,y\right)$ be the midpoint.

We know that the midpoint between two points $A\left(x_1,y_1\right)andB\left(x_2,y_2\right)$ is given by

  $P\left(x,y\right)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Now, using the distance formula we get

  $\begin{array}{c}P\left(x,y\right)=\left(\frac{1+9}{2},\frac{1+7}{2}\right)\\ =\left(\frac{10}{2},\frac{8}{2}\right)\\ =\left(5,4\right)\end{array}$

Therefore, the midpoint between given points is $P\left(x,y\right)=\left(5,4\right)$