Q. 4 - 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.  (0, 0) and (3, 4)

Step-by-Step Solution

Answer

The distance is 5.

Step-1 – Given

The given points are (0, 0) and (3, 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 $A\left(x_1,y_1\right)\text{ is }A\left(0,0\right)\text{ and }B\left(x_2,y_2\right)\text{ is }B\left(3,4\right)$

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

So, the coordinate of T is $\left(x_2,y_1\right)=\left(3,0\right)$

Then,

$$\begin{gathered} \begin{array}{l}AT=\left(x_1-x_2\right)\\ AT=\left(0-3\right)\\ AT=\left(-3\right)\\ AT=-3\end{array} \\ And \end{gathered}$$

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

Distances can not be negative. So, AT = 3 and BT = 4.

Using the Pythagorean theorem, we get:
$\begin{array}{l}{\left(AB\right)}^2={\left(AT\right)}^2+{\left(BT\right)}^2\\ {\left(AB\right)}^2={\left(3\right)}^2+{\left(4\right)}^2\\ {\left(AB\right)}^2=9+16\\ {\left(AB\right)}^2=25\\ AB=5\end{array}$

Hence, the required distance is 5.