Question

How can the distance formula be derived from the pythagorean theorem?

Answer 1

Let's see.

Explanation:

I have drawn a graph in which there are two points `color(red)(p_1(x_1,y_1))" and "color(red)(p_2(x_2,y_2)`.

• We can easily say that

  `" "bar(OD)=x_1" ; "bar(OE)=x_2" ; "bar(AD)=y_1" ; "bar(EB)=y_2`

We also have a rectangle `square OCED`. So, `color(red)(bar(AC)=bar(DE)) " and "color(red)(bar(AD)=bar(CE)`

Now,

• `bar(AC)=bar(DE)=bar(OE)-bar(OD)=(x_2 -x_1)`

• `bar(BC)=bar(BE)-bar(CE)=bar(BE)-bar(AD)=(y_2-y_1)`

With the help of Pythagorean theorem,

`bar(AB)^2=bar(BC)^2+bar(AC)^2`

`bar(AB)^2=(x_2-x_1)^2+(y_2-y_1)^2`

`bar(AB)=sqrt((x_2-x_1)^2+(y_2-y_1)^2`

N.B:- As it is a square value , you may take `(x_1-x_2)` or, `(x_2-x_1)`. I mean you have to take difference.That's `(x_1~x_2)`

So, the required formula is proved that

If the distance between two points `color(green)(p_1(x_1,y_1)` and `color(green)(p_2(x_2,y_2)` is `color(red)(r`,

then, `color(red)(ul(bar(|color(green)(r=sqrt((x_1-x_2)^2+(y_1-y_2)^2))|`

Hope it helps...
Thank you...