Chapter 1.8, Problem 12E

  $\left(a\right)$

To determine

To calculate: The distance between the pair of points that are graphed below.

  

Answer to Problem 12E

The distance between the pair of points is $2\sqrt{10}$ units

Explanation of Solution

Given information:

The points are depicted graphically,

  

Formula used:

Distance $d\left(A,B\right)$ between two points $A=\left(x_1,y_1\right)\text{ and }B=\left(x_2,y_2\right)$ in the Cartesian plane is denoted by $d\left(A,B\right)=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$ .

Calculation:

Consider the points that are depicted graphically,

  

The two points are $A\left(4,-1\right)$ and $B\left(-2,-3\right)$ .

Recall that the distance $d\left(A,B\right)$ between two points $A=\left(x_1,y_1\right)\text{ and }B=\left(x_2,y_2\right)$ in the Cartesian plane is denoted by $d\left(A,B\right)=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$ .

Evaluate the distance between $A\left(4,-1\right)$ and $B\left(-2,-3\right)$ .

  $\begin{array}{c}d\left(A,B\right)=\sqrt{{\left(-2-4\right)}^2+{\left(-3-\left(-1\right)\right)}^2}\\ =\sqrt{36+4}\\ =\sqrt{40}\\ =2\sqrt{10}\end{array}$

Thus, the distance between the pair of points is $2\sqrt{10}$ units.

  $\left(b\right)$

To determine

To calculate: The midpoint of the segment that joins the pair of points that are graphed below.

  

Answer to Problem 12E

The midpoint of the segment that joins the pair of points is $\left(1,-2\right)$ .

Explanation of Solution

Given information:

The points are depicted graphically,

  

Formula used:

Midpoint of the segment that joins two points $A=\left(x_1,y_1\right)\text{ and }B=\left(x_2,y_2\right)$ in the Cartesian plane is denoted by $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$ .

Calculation:

Consider the points that are depicted graphically,

  

The two points are $A\left(4,-1\right)$ and $B\left(-2,-3\right)$ .

Recall that the midpoint of the segment that joins two points $A=\left(x_1,y_1\right)\text{ and }B=\left(x_2,y_2\right)$ in the Cartesian plane is denoted by $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$ .

Evaluate the midpoint of the segment that joins $A\left(4,-1\right)$ and $B\left(-2,-3\right)$ .

  $\begin{array}{c}\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)=\left(\frac{4+\left(-2\right)}{2},\frac{-1+\left(-3\right)}{2}\right)\\ =\left(\frac{2}{2},\frac{-4}{2}\right)\\ =\left(1,-2\right)\end{array}$

Thus, the midpoint of the segment that joins the pair of points is $\left(1,-2\right)$ .