Chapter 9.5, Problem 41E

To determine

Toexplain:If replacing $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ changes the final value of distance formula

Explanation of Solution

Concept Used:

Distance between two points - $d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

We know-

  ${\left(a-b\right)}^2={\left(b-a\right)}^2$

For example-

  $\begin{array}{l}a=4\text{ and }b=-3\\ {\left(4-\left(-3\right)\right)}^2={\left(-4-3\right)}^2\\ {\left(7\right)}^2={\left(-7\right)}^2\\ 49=49\end{array}$

Similarly-

  ${\left(x_2-x_1\right)}^2={\left(x_1-x_2\right)}^2$

Therefore,

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

Hence, there will be no change in the distance.