Chapter 1.1, Problem 36E

a)

To determine

Tograph : the given points in the coordinate axis.

Explanation of Solution

Given information : we have the given points are $\left(\frac{1}{2},1\right)and\left(\frac{-5}{2},\frac{4}{3}\right)$

Graph :

The plotting of the points is shown in the graph below

  

Interpretation :

The point $\left(\frac{1}{2},1\right)$ is shown by blue dot and point $\left(\frac{-5}{2},\frac{4}{3}\right)$ is shown by red dot.

b)

To determine

To find : the distance between two given points.

Answer to Problem 36E

The distance between given points is $\sqrt{\frac{82}{9}}$ units.

Explanation of Solution

Given information : we have the given points are $\left(\frac{1}{2},1\right)and\left(\frac{-5}{2},\frac{4}{3}\right)$

Let the points be $A\left(\frac{1}{2},1\right)andB\left(\frac{-5}{2},\frac{4}{3}\right)$

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

  $\begin{array}{c}AB=\sqrt{{\left(\frac{1}{2}+\frac{5}{2}\right)}^2+{\left(1-\frac{4}{3}\right)}^2}\\ =\sqrt{{\left(3\right)}^2+{\left(\frac{-1}{3}\right)}^2}\\ =\sqrt{9+\frac{1}{9}}\\ =\sqrt{\frac{82}{9}}\end{array}$

Therefore, the distance between given points is $\sqrt{\frac{82}{9}}$ units.

c)

To determine

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

Answer to Problem 36E

The midpoint between given points is $P\left(x,y\right)=\left(-1,\frac{7}{6}\right)$

Explanation of Solution

Given information : we have the given points are $\left(\frac{1}{2},1\right)and\left(\frac{-5}{2},\frac{4}{3}\right)$

Let the points be $A\left(\frac{1}{2},1\right)andB\left(\frac{-5}{2},\frac{4}{3}\right)$ 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

  $\begin{array}{c}P\left(x,y\right)=\left(\frac{\frac{1}{2}-\frac{5}{2}}{2},\frac{1+\frac{4}{3}}{2}\right)\\ =\left(-1,\frac{7}{6}\right)\end{array}$

Therefore, the midpoint between given points is $P\left(x,y\right)=\left(-1,\frac{7}{6}\right)$