Q. 3 - Geometry

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.  (3, -4) and (-1, -4)

Step-by-Step Solution

Answer

The distance is 4.

Step-1 – Given

The given points are (3, -4) and (-1, -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 $A\left(x_1,y_1\right)\text{ be }A\left(3,-4\right)\text{ and }B\left(x_2,y_2\right)\text{ be }B\left(-1,-4\right)$

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

So, the coordinate of T is $\left(x_2,y_1\right)=\left(-1,-4\right)$

Then,

$$\begin{gathered} \begin{array}{l}AT=\left(x_1-x_2\right)\\ AT=\left(3-\left(-1\right)\right)\\ AT=\left(3+1\right)\\ AT=4\end{array} \\ And \end{gathered}$$

$\begin{array}{l}BT=\left(y_1-y_2\right)\\ BT=\left(-4-\left(-4\right)\right)\\ BT=\left(-4+4\right)\\ BT=0\end{array}$

Using the Pythagorean theorem, we get:

$\begin{array}{l}{\left(AB\right)}^2={\left(AT\right)}^2+{\left(BT\right)}^2\\ {\left(AB\right)}^2={\left(4\right)}^2+{\left(0\right)}^2\\ {\left(AB\right)}^2=16+0\\ {\left(AB\right)}^2=16\\ AB=4\end{array}$

Hence, the required distance is 4.