Use Distance Formula to Find the Length of a Line

You can measure the length of a vertical or horizontal line on a coordinate plane by simply counting coordinates; however, measuring the length of a diagonal line is trickier. You can use the Distance Formula to find the length of such a line. This formula is basically the Pythagorean Theorem, which you can see if you imagine the given line segment as the hypotenuse of a right triangle.[1] By using a basic geometric formula, measuring lines on a coordinate path becomes a relatively easy task.

Steps

Setting up the Formula

  1. Set up the Distance Formula. The formula states that <math>d = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}</math>, where <math>d</math> equals the distance of the line, <math>(x_{1}, y_{1})</math> equal the coordinates of the first endpoint of the line segment, and <math>(x_{2}, y_{2})</math> equal the coordinates of the second endpoint of the line segment.[2]
  2. Find the coordinates of the line segment’s endpoints. These might already be given. If not, count along the x-axis and y-axis to find the coordinates.
    • The x-axis is the horizontal axis; the y-axis is the vertical axis.
    • The coordinates of a point are written as <math>(x, y)</math>.
    • For example, a line segment might have an endpoint at <math>(2, 1)</math> and another at <math>(6, 4)</math>.
  3. Plug the coordinates into the Distance Formula. Be careful to substitute the values for the correct variables. The two <math>x</math> coordinates should be inside the first set of parentheses, and the two <math>y</math> coordinates should be inside the second set of parentheses.
    • For example, for points <math>(2, 1)</math> and <math>(6, 4)</math>, your formula would look like this: <math>d = \sqrt{(6 - 2)^{2} + (4 - 1)^{2}}</math>

Calculating the Distance

  1. Calculate the subtraction in parentheses. By using the order of operations, any calculations in parentheses must be completed first.
    • For example:
      <math>d = \sqrt{(6 - 2)^{2} + (4 - 1)^{2}}</math>
      <math>d = \sqrt{(4)^{2} + (3)^{2}}</math>
  2. Square the value in parentheses. The order of operations states that exponents should be addressed next.
    • For example:
      <math>d = \sqrt{(4)^{2} + (3)^{2}}</math>
      <math>d = \sqrt{16 + 9}</math>
  3. Add the numbers under the radical sign. You do this calculation as if you were working with whole numbers.
    • For example:
      <math>d = \sqrt{16 + 9}</math>
      <math>d = \sqrt{25}</math>
  4. Solve for <math>d</math>. To reach your final answer, find the square root of the sum under the radical sign.
    • Since you are finding a square root, you may have to round your answer.
    • Since you are working on a coordinate plane, your answer will be in generic “units,” not in centimeters, meters, or another metric unit.
    • For example:
      <math>d = \sqrt{25}</math>
      <math>d = 5</math> units

Tips

  • Do not confuse this formula with others, like the Midpoint Formula, Slope Formula, Equation of a Line or Line Formula.
  • Remember the order of operations when calculating your answer. Subtract first, then square the differences, then add, and then find the square root.

Related Articles

Sources and Citations

  1. ↑ http://www.purplemath.com/modules/distform.htm
  2. ↑ http://mathworld.wolfram.com/Distance.html
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