Question

Find the distance between the two points. If necessary, you may draw graphs but you shouldn’t need to use the distance formula.

1. (2, -3) and (2, 4)

Short answer

The distance is 7.

Step-by-step solution

Step-1 – Given

The given points are (-2, -3) and (-2, 4).

Step-2 – To determine

We have to find the distance between the two points without using distance formula.

Step-3 – Calculation 

Let, the coordinate of$A\left(x_1,y_1\right)\text{is}A\left(-2,-3\right)\text{and}B\left(x_2,y_2\right)\text{is}B\left(-2,4\right)$

Let T be the point where the horizontal line through A and vertical line through B meets.

So, the coordinate of Tis$\left(x_2,y_1\right)=\left(-2,-3\right)$

Then,

$\begin{array}{l}AT=\left(x_1-x_2\right)\\ AT=\left(-2-\left(-2\right)\right)\\ AT=\left(-2+2\right)\\ AT=0\end{array}$

And

$\begin{array}{l}BT=\left(y_1-y_2\right)\\ BT=\left(-3-4\right)\\ BT=\left(-7\right)\\ BT=-7\end{array}$

BT is a distance and distancecannot be negative. Therefore, BT = 7.

Using the Pythagorean theorem we get:

$\begin{array}{l}{\left(AB\right)}^2={\left(AT\right)}^2+{\left(BT\right)}^2\\ {\left(AB\right)}^2={\left(0\right)}^2+{\left(7\right)}^2\\ {\left(AB\right)}^2=0+49\\ {\left(AB\right)}^2=49\\ AB=7\end{array}$

Hence, the required distance is 7.