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Related Lessons: Midpoint Formula Derivation of Distance Formula

 

The Distance Formula

 

The distance formula is a useful tool in finding the distance between two points. The formula itself is actually derived from the Pythagorean Theorem. That's why we can claim that the distance formula is simply the Pythagorean Theorem in disguise.

If you want to see how the Distance Formula is derived from the Pythagorean Theorem, check it out on my separate lesson.

 

Distance Formula

The distance d between two points

  • point A with coordinates (x1,y1) and point B with coordinates (x2,y2)

is calculated as follows:

  • d = sqrt [ (x2-x1)^2 + (y2-y1)^2 ]
diagram of distance formula using two arbitrary points

Observations:

a) The expression x2-x1 is read as "the change in x".

b) The expression y2-y1 is read as "the change in y".

 


Example 1: Find the distance between the points (-3,2) and (3,5).

Label the parts of each point properly and substitute into the distance formula.

If we let (-3,2) be the first point then it will take the subscript of 1, thus , x1=-3 and y1=2. Similarly, if (3,5)be the second point it will have the subscript of 2, thus, x2=3 and y2=5.

Here is the calculation,

shows the calculation of the distance between points (-3,2) and (3,5) which is d=3*sqrt(5)

This is how it looks in the graph.

diagram of the distance between points (-3,2) and (3,5)

 


Example 2: Find the distance between the points (-1,-1) and (4,-5).

If we assign (-1,-1)as our first point then x1=-1 and y1=-1. In the same manner, assigning (4,-5) as our second point we have x2=4 and y2=-5.

Plugging the values of x and y in the distance formula, we get...

distance = sqrt (41)

The two points and the distance between them looks like this in the graph.

diagram of the distance between points (-1,-1) and (4,-5)

 


Example 3: Find the distance between the points (-4,-3) and (4,3).

Sometimes you may wonder if switching the points in calculating the distance can affect the final outcome.

Well, if you think about, the formula is squaring the difference of the corresponding x and y values. That means, it doesn't matter if the change in x or change in y is negative because when we eventually square it (raise to the 2nd power), the result always comes out to be positive.

Let's "prove" that the answer is always the same by solving this problem two ways!

The first solution shows the usual way because we assign which point is the first and second based on the order in which they are given to us in the problem. In the second solution, we switch the points.

 

Solution 1 Solution 2
first point is (-4,-3), and the second point is (4,3) the first point is (4,3) and the second point is (-4,-3)
shows the calculation of distance d=10

As you can see, both answers came out the same which is d = 10. Below is the visual solution to the problem.

image illustrating how the distance if calculated between points (-4,-3) and (4,3)

 


Example 4: Find the radius of a circle with a diameter whose endpoints are (-7,1) and (1,3).

Remember that the diameter of a circle is twice the length of its radius. If that's the case, then the radius is half the length of the diameter.

radius = half of diameter

 

Here's the plan! Since we are given with the endpoints of the diameter, we can use the distance formula to find its length. Finally, we divide it by 2 to get the length of the radius, as required by the problem.

  • Find the length of the diameter with endpoints (-7,1) and (1,3) using the distance formula.
distance = 2*sqrt(17)
  • Solve for the radius by dividing the diameter by 2.
radius = (1/2)*(2 sqrt(17)) = sqrt(17)
  • The blue dots are the endpoints of diameter and the green dot is the center of the circle located at (-3,2).
image of circle showing the radius is equal to sqrt (17) as calculated using the distance formula

 


 

Practice Problems with Answers
Worksheet 1 Worksheet 2

 

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