Q 1 - 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). 

$\left(0,0\right);\left(4,2\right)$.

Step-by-Step Solution

Answer

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

 $2\sqrt{5}$.

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

$M(2,1)$.

(c) The slope of the line containing given points is :-

 $m=\frac{1}{2}$.

(d) If $x$ increases by $2$ units, then $y$ will increases by $1$ unit.

Step 1. Given Information

We have give the following pair of points :-

$\left(0,0\right);\left(4,2\right)$.

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 :-

$\left(0,0\right);\left(4,2\right)$.

Name these as following :-

$A\left(0,0\right);B\left(4,2\right)$.

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 :-

$$\begin{gathered} d\left(A,B\right)=\sqrt{{\left(4-0\right)}^2+{\left(2-0\right)}^2} \\ \Rightarrow d\left(A,B\right)=\sqrt{16+4} \\ \Rightarrow d\left(A,B\right)=\sqrt{20} \\ \Rightarrow d\left(A,B\right)=2\sqrt{5} \end{gathered}$$

So the distance between the given points is $2\sqrt{5}$.

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

The given points are :-

$A\left(0,0\right);B\left(4,2\right)$.

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

$$\begin{gathered} M\left(x,y\right)=\left(\frac{0+4}{2},\frac{0+2}{2}\right) \\ \Rightarrow M(x,y)=\left(\frac{4}{2},\frac{2}{2}\right) \\ \Rightarrow M(x,y)=(2,1) \end{gathered}$$

So the midpoint of the line segment joining given two points $A(0,0) and B(4,2)$ is $M\left(2,1\right)$.

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

The given two points are :-

$A\left(0,0\right);B(4,2)$.

We know that slope $m$ of a non vertical 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 :-

$$\begin{gathered} m=\frac{2-0}{4-0} \\ \Rightarrow m=\frac{2}{4} \\ \Rightarrow m=\frac{1}{2} \end{gathered}$$

So the slope of the line connecting through given two points $A\left(0,0\right) and B\left(4,2\right)$ is  $m=\frac{1}{2}$.

Step 5. Part (d) Interpretation of slope

In part (c) we find that slope of the line containing  give two points $A\left(0,0\right) and B(4,2)$ is $m=\frac{1}{2}$.

That is for every $2-$units change in $x$, $y$ will change by $1$ unit. That is if $x$ increases by $2$ units, the $y$ will increases by $1$ unit. The average rate of change of $y$ with respect to $x$ is $\frac{1}{2}$.