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The Distance Formula (page 2 of 2)


The most common mistake made when using the Formula is to accidentally mismatch the x-values and y-values. Be careful you don't subtract an x from a y, or vice versa; make sure you've paired the numbers properly.

 

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Also, don't get careless with the square-root symbol. If you get in the habit of omitting the square root and then "remembering" to put it back in when you check your answers in the back of the book, then you'll forget the square root on the test, and you'll miss easy points.

You also don't want to be careless with the squaring inside the Formula. Remember that you simplify inside the parentheses before you square, not after, and remember that the square is on everything inside the parentheses, including the minus sign, so the square of a negative is a positive.

By the way, it is almost always better to leave the answer in "exact" form (the square root "sqrt(53)" above). Rounding is usually reserved for the last step of word problems. If you're not sure which format is preferred, do both, like this:

    d = sqrt(53), or about 7.28

Now YOU try it!


Very often you will encounter the Distance Formula in veiled forms. That is, the exercise will not explicitly state that you need to use the Distance Formula; instead, you have to notice that you need to find the distance, and then remember (and apply) the Formula. For instance:

  • Find the radius of a circle, given that the center is at (2, 3) and the point (1, 2) lies on the circle.

    The radius is the distance between the center and any point on the circle, so I need to find the distance: Copyright Elizabeth Stapel 2000-2011 All Rights Reserved

      d = sqrt[ (2 + 1)^2 + (-3 + 2)^2 ] = sqrt(10)

    Then the radius is sqrt(10), or about 3.16, rounded to two decimal places.

  • Find all points (4, y) that are 10 units from the point (2, 1).
  • I'll plug the two points and the distance into the Distance Formula:

      10 = sqrt[(-2 - 4)^2 + (-1 - y)^2] = [y^2 + 2y + 37]

    Now I'll square both sides, so I can get to the variable:

      100 = y^2 + 2y + 37, so 0 = y^2 + 2y - 63

    This means y = 9 or y = 7, so the two points are (4, 9) and (4, 7).

If you're not sure why there are two points that solve this exercise, try drawing the (2, 1) and then drawing a circle with radius 10 around this. Then draw the vertical line through x = 4. You'll see that the vertical line crosses the circle in two spots: (4, 9) and (4, 7).


You can use the Mathway widget below to practice finding the distance between two points. Try the entered exercise, or type in your own exercise. Then click "Answer" to compare your answer to Mathway's.

(Clicking on "View Steps" on the widget's answer screen will take you to the Mathway site, where you can register for a free seven-day trial of the software.)

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Cite this article as:

Stapel, Elizabeth. "The Distance Formula: Worked Examples." Purplemath. Available from
    http://www.purplemath.com/modules/distform2.htm. Accessed
 

 



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