Standards:
MCC912.G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. ^{ ★ }
DISTANCE FORUMLA
What if you were given the coordinates of two points like (6, 2) and (3, 0). How could you determine how far apart these two points are? After completing this Concept, you'll be able to find the distance between any two points in the coordinate plane using the Distance Formula.
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CK12 Foundation: The Distance Formula
Guidance
In the last section, we saw how to use the Pythagorean Theorem to find lengths. In this section, you’ll learn how to use the Pythagorean Theorem to find the distance between two coordinate points.
Example A
Find the distance between points
Solution
Plot the two points on the coordinate plane.
In order to get from point
To find the distance between
Example B
Find the distance between points
Solution
We plot the two points on the graph above.
In order to get from point
We find the distance from
The Distance Formula
The procedure we just used can be generalized by using the Pythagorean Theorem to derive a formula for the distance between any two points on the coordinate plane.
Let’s find the distance between two general points
Start by plotting the points on the coordinate plane:
In order to move from point
We can find the length
Therefore,
Given any two points
We can use this formula to find the distance between any two points on the coordinate plane. Notice that the distance is the same whether you are going from point
Let’s now apply the distance formula to the following examples.
Example C
Find the distance between the following points.
a) (3, 5) and (4, 2)
b) (12, 16) and (19, 21)
c) (11.5, 2.3) and (4.2, 3.9)
Solution
Plug the values of the two points into the distance formula. Be sure to simplify if possible.
a)
b)
c)
Applications Using Distance and Midpoint Formulas
The distance and midpoint formula are useful in geometry situations where we want to find the distance between two points or the point halfway between two points.
Example D
Plot the points
Solution
Let’s start by plotting the three points on the coordinate plane and making a triangle:
We use the distance formula three times to find the lengths of the three sides of the triangle.
Notice that
Example E
At 8 AM one day, Amir decides to walk in a straight line on the beach. After two hours of making no turns and traveling at a steady rate, Amir is two miles east and four miles north of his starting point. How far did Amir walk and what was his walking speed?
Solution
Let’s start by plotting Amir’s route on a coordinate graph. We can place his starting point at the origin:
The distance can be found with the distance formula:
Since Amir walked 4.47 miles in 2 hours, his speed is
Watch this video for help with the Examples above.
CK12 Foundation: The Distance Formula
Vocabulary

The
Distance Formula
states that given any two points
(x1,y1) and(x2,y2) , the distance between them is
Guided Practice
Find all points on the line
Solution
Let’s make a sketch of the given situation.
Draw line segments from the point (3, 7) to the line
Let
The points are (9.24, 2) and (3.24, 2).
Practice
Find the distance between the two points.
 (3, 4) and (6, 0)
 (1, 0) and (4, 2)
 (3, 2) and (6, 2)
 (0.5, 2.5) and (4, 4)
 (12, 10) and (0, 6)
 (5, 3) and (2, 11)
 (2.3, 4.5) and (3.4, 5.2)

Find all points having an
x− coordinate of 4 whose distance from the point (4, 2) is 10. 
Find all points having a
y− coordinate of 3 whose distance from the point (2, 5) is 8. 
Find three points that are each 13 units away from the point (3, 2) but do
not
have an
x− coordinate of 3 or ay− coordinate of 2.
Find the midpoint of the line segment joining the two points.

Plot the points
A=(1,0),B=(6,4),C=(9,−2) andD=(−6,−4),E=(−1,0),F=(2,−6) . Prove that trianglesABC andDEF are congruent. 
Plot the points
A=(4,−3),B=(3,4),C=(−2,−1),D=(−1,−8). Show thatABCD is a rhombus (all sides are equal) 
Plot points
A=(−5,3),B=(6,0),C=(5,5). Find the length of each side. Show thatABC is a right triangle. Find its area.  Find the area of the circle with center (5, 4) and the point on the circle (3, 2).
 Michelle decides to ride her bike one day. First she rides her bike due south for 12 miles and then the direction of the bike trail changes and she rides in the new direction for a while longer. When she stops Michelle is 2 miles south and 10 miles west from her starting point. Find the total distance that Michelle covered from her starting point.