Q 3 - 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). 

$\left(4,-4\right);\left(4,8\right)$.

Step-by-Step Solution

Answer

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

$12$.

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

$M(4,2)$.

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

(d) There is no change in $x$.

Step 1. Given Information

We have give the following pair of points :- 

$\left(4,-4\right);\left(4,8\right)$.

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 :- 

$\left(4,-4\right);\left(4,8\right)$.

Name these as following :- 

$A(4,-4);B(4,8)$.

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

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

So that distance between given points is :- 

$$\begin{gathered} d(A,B)=\sqrt{{\left(4-4\right)}^2+{\left(8+4\right)}^2} \\ \Rightarrow d\left(A,B\right)=\sqrt{0+{12}^2} \\ \Rightarrow d(A,B)=\sqrt{144} \\ \Rightarrow d(A,B)=12 \end{gathered}$$

So the distance between the given points is $12$.

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

The given points are :- 

$A\left(4,-4\right) and B\left(4,8\right)$.

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

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

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

$$\begin{gathered} M(x,y)=\left(\frac{4+4}{2},\frac{8-4}{2}\right) \\ \Rightarrow M(x,y)=\left(\frac{8}{2},\frac{4}{2}\right) \\ \Rightarrow M(x,y)=\left(4,2\right) \end{gathered}$$

So the midpoint of the line segment joining given two points $A\left(4,-4\right) and B(4,8)$ is $M(4,2)$.

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

The given two points are :- 

$A(4,-4) and B(4,8)$.

We know that slope $m$ of a line connecting through two points $P_1(x_1,y_1);P_2(x_2,y_2)$ is $m=\frac{y_2-y_1}{x_2-x_1}$.

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

$$\begin{gathered} m=\frac{8+4}{4-4} \\ \Rightarrow m=\frac{12}{0} \end{gathered}$$

As there is denominator is zero. So the slope of the line containing given two points is undefined as the line containing these points is a vertical line.

Step 5. Interpretation of slope

As we find slope of given line is undefined.

Also we have two points $A\left(4,-4\right) and B\left(4,8\right)$ connecting a line.

In these two points the value of $x$ is same. So the value of data-custom-editor="chemistry" $\mathrm{x}$ will be same for all points lie on a line.

So we can conclude that there is no change in $x$.