Question

In Exercises 21-24. A pair of points is graphed.

a. Find the distance between them.

b. Find the midpoint of the segment that joins them.

Short answer

1. The distance between the points is 5 units.
2. The midpoint of the segment is $\left(0,\frac{1}{2}\right)$.

Step-by-step solution

a.Step 1. Apply the concept of distance formula

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

Step 2. Find coordinates from graph

Consider the points that are depicted graphically.

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

Step 3. Compute the distance 

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

Apply it. Compute the distance between the points $A\left(-2,-1\right)$ and $B\left(2,2\right)$.

$\begin{array}{c}d\left(A,B\right)=\sqrt{{\left(2-\left(-2\right)\right)}^2+{\left(2-\left(-1\right)\right)}^2}\\ =\sqrt{4^2+3^2}\\ =\sqrt{16+9}\\ =\sqrt{25}\\ =5\end{array}$

Thus, the distance between the points $A\left(-2,-1\right)$ and $B\left(2,2\right)$ is 5 units.

b.Step 1. Apply the concept of midpoint formula

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

Step 2. Find coordinates from graph

Consider the points that are depicted graphically.

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

Step 3. Compute the midpoint 

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

Apply it. Compute the midpoint of the points $A\left(-2,-1\right)$ and $B\left(2,2\right)$.

$\begin{array}{c}\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)=\left(\frac{-2+2}{2},\frac{-1+2}{2}\right)\\ =\left(0,\frac{1}{2}\right)\end{array}$

Thus, the midpoint of line segment that join the points $A\left(-2,-1\right)$ and $B\left(2,2\right)$ is $\left(0,\frac{1}{2}\right)$.