Distance Formula

Key Questions

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

    Answer:

    Let's see.

    Explanation:

    ![my http://notebook...](https://useruploads.socratic.org/78ifMgjvSTyOjVwe4b02_IMG_20180406_213532%5B1%5D.jpg)

    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...

    Sunayan S · 1 · Apr 6 2018
  • How do you find the distance when given two coordinate points?

    Distance Formula

    The distance D between two points (x_1,y_1) and (x_2,y_2) can be found by

    D=sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

    Example

    Find the distance between the points (1,-2) and (5,1).

    Let (x_1,y_1)=(1,-2) and (x_2,y_2)=(5,1).

    By Distance Formula above,

    D=sqrt{(5-1)^2+[1-(-2)]^2}=sqrt{16+9}=sqrt{25}=5

    I hope that this was helpful.

    Wataru · 1 · Nov 6 2014
  • What is the Distance Formula?

    Given the two points (x_1, y_1) and (x_2, y_2) the distance between these points is given by the formula:

    d=sqrt((x_2-x_1)^2+(y_2-y_1)^2 )

    All you have to do is plug in the given points given to the distance formula and solve.

    Mika · 1 · Dec 18 2014

Questions

Radicals and Geometry Connections