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Distance Formula

The distance formula to find the distance between two points (x1, y1) and (x2, y2) is derived from the Pythagorean Theorem.

Let A and B be two points on the coordinate plane, with coordinates (x1, y1) and (x2, y2) respectively. See the figure below:

GRAPH

Then by Pythagorean Theorem, AB2\sqrt{ \left ( x_{2} -x_{1}\right )^{2}+ \left ( y_{2} -y_{1}\right )^{2}}

Example: Find the distance between the points (8, -4) and (8, -7).

Let us assume (x1, y1) = (8, -4) and (x2, y2) = (8, -7)

Then distance between the given points = \sqrt{ \left ( x_{2} -x_{1}\right )^{2}+ \left ( y_{2} -y_{1}\right )^{2}}

=\sqrt{ \left (8 -8\right )^{2}+ \left ( -7 +4\right )^{2}} =  \sqrt{ \left (0 )^{2}+ \left ( -3\right )^{2}} = \sqrt{9} = 3 units

 

Example: What is the distance between the points (-6, -1) and (2, 5)?

Let us assume (x1, y1) = (-6, -1) and (x2, y2) = (2, 5)

Then distance between the given points = \sqrt{\left ( x_{2} -x_{1}\right )^{2}+ \left ( y_{2} -y_{1}\right )^{2}}

= \sqrt{\left ( 2 -(-6)\right )^{2} + \left ( 5 -(-1)\right )^{2}} = \sqrt{\left ( 2 + 6\right )^{2} + \left ( 5 +1\right )^{2}}

\sqrt{ 8^{2}+ 6^{2}} = \sqrt{ 64+ 36} = \sqrt{ 100} = 10 units

distance1-300x241

 

Find the distance between each pair of points.

  1. (-2, -2), (-5, -6)
  2. (-4, 0), (-7, 0)
  3. (-1, 6), (7, 0)
  4. (2, -3), (3, -5)
  5. (-8, 1), (7, 1)
Answer key
  1. 5 units
  2. 3 units
  3. 10 units
  4. \sqrt{ 5} units
  5. 15 units
Download/Solve a Worksheet for Distance between points

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