Chapter 1, Problem 103RE

To determine

To calculate:Among the two given points $A\left(4,4\right)$ and $B(5,3)$ which one of them is closer to the third point $C(-1,-3)$ .

Answer to Problem 103RE

Point $B(5,3)$ is closer to point $C(-1,-3)$ .

Explanation of Solution

Given information:

The three points $A\left(4,4\right)$ , $B(5,3)$ and $C(-1,-3)$

Formula used:

The Distance denoted by say, $D$ between any two given points say $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is given as:

  $D=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Calculation:

Consider the points $A\left(4,4\right)$ and $C(-1,-3)$

Then in order to apply the distance formula the following should be known:

  $\begin{array}{l}{x}_1=4\\ x_2=4\\ y_1=-1\\ y_2=-3\end{array}$

Recall that for two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is given as:

  $D=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Therefore, if $D_1$ denotes the distance between $A\left(4,4\right)$ and $C(-1,-3)$ then it is given as follows:

  $\begin{array}{l}{D}_1=\sqrt{{\left(-1-4\right)}^2+{\left(-3-4\right)}^2}\\ =\sqrt{{\left(-5\right)}^2+{\left(-7\right)}^2}\\ =\sqrt{25+49}\\ =\sqrt{74}\end{array}$

Now, consider the points $B(5,3)$ and $C(-1,-3)$

Then in order to apply the distance formula the following should be known:

  $\begin{array}{l}{x}_1=5\\ x_2=3\\ y_1=-1\\ y_2=-3\end{array}$

Recall that for two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is given as:

  $D=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Therefore, if $D_2$ denotes the distance between $B(5,3)$ and $C(-1,-3)$ then it is given as follows:

  $\begin{array}{l}{D}_2=\sqrt{{\left(-1-5\right)}^2+{\left(-3-3\right)}^2}\\ =\sqrt{{\left(-6\right)}^2+{\left(-6\right)}^2}\\ =\sqrt{36+36}\\ =\sqrt{72}\end{array}$

Clearly as $\sqrt{74}\succ \sqrt{72}$

That is $D_1\succ D_2$

Using the fact that lesser the distance, closer the point.

Hence, $B(5,3)$ is closer to $C(-1,-3)$