Distance Formula

Key Questions

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

    Answer:

    Let's see.

    Explanation:

    my notebook...

    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