Q 4 - Precalculus Enhanced with Graphing Utilities

Question

Find the following for each pair of points:

(a) The distance between the points.

(b) The midpoint of the line segment connecting the points.

(c) The slope of the line containing the points.

(d) Then interpret the slope found in part (c). 

$\left(-2,-1\right);\left(3,-1\right)$.

Step-by-Step Solution

Answer

(a) The distance between given two points is :-

$5$.

(b) The midpoint of the line segment connecting the given points is :- 

$M\left(\frac{1}{2},-1\right)$.

(c) The slope of the line containing given points is $0$.

(d) There is no change in $y$.

Step 1. Given Information

We have give the following pair of points :- 

$\left(-2,-1\right);\left(3,-1\right)$.

We have to find the distance between these two points, mid point of the line segment connecting these points and slope of the line connecting these points. Also we have to interpret the slope.  

Step 2. Part (a) Distance between the points.

We have given the following two points :-  

$\left(-2,-1\right);(3,-1)$.

Name these as following :- 

$A\left(-2,-1\right);B\left(3,-1\right)$.

We know that distance between two points $P_1(x_1,y_1) and P_2(x_2,y_2)$ is defined as :-

$d(P_1,P_2)=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

So that distance between given points is :- 

$$\begin{gathered} d(A,B)=\sqrt{{\left(3+2\right)}^2+{\left(-1+1\right)}^2} \\ \Rightarrow d(A,B)=\sqrt{25+0} \\ \Rightarrow d(A,B)=\sqrt{25} \\ \Rightarrow d(A,B)=5 \end{gathered}$$

So the distance between the given points is  $5$.

Step 3. Part (b) Midpoint of the line segment joining given points.

The given points are :-  

$A\left(-2,-1\right);B(3,-1)$.

We know that the midpoint $M\left(x,y\right)$ of the line segment from   $P_1(x_1,y_1) to P_2(x_2,y_2)$ is :-

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

So the midpoints of the line segment joining given two points is :-  

$$\begin{gathered} M(x,y)=\left(\frac{-2+3}{2},\frac{-1-1}{2}\right) \\ \Rightarrow M(x,y)=\left(\frac{1}{2},-\frac{2}{2}\right) \\ \Rightarrow M(x,y)=\left(\frac{1}{2},-1\right) \end{gathered}$$

So the midpoint of the line segment joining given two points $A(-2,-1) and B\left(3,-1\right)$ is  $M\left(\frac{1}{2},-1\right)$.

Step 4. Part (c) Slope of the line containing the points

The given two points are :-  

$A\left(-2,-1\right);B(3,-1)$.

We know that slope $m$ of a line connecting through two points $P_1(x_1,y_1) and P_2(x_2,y_2)$ is :-

$m=\frac{y_2-y_1}{x_2-x_1}$.

So the slope of the line connecting through given two points is :-  

$$\begin{gathered} m=\frac{-1+1}{3+2} \\ \Rightarrow m=\frac{0}{5} \\ \Rightarrow m=0 \end{gathered}$$

That is slope of the line connecting given two points is $0$. We can say it is parallel to $x$ axis.

Step 5. Interpretation of slope

As we find slope of given line is $0$.

That is there is no change in $y$, when there is any change in $x$.

That is $y$ remains same for every point.

So The line is a horizontal line that containing given two points.