the distance formula.

1. The
Distance
Formula

2. Table of contents
01
02
Recap on slope and
Introduction to distance
The distance formula and how
it is constructed and used

3. Recap on slope
and introduction
to distance
An introduction!
01

4. An introduction!
In algebra and geometry, we usually
work with the coordinate plane,
which is composed of the x
(horizontal) and y (vertical) axes. We
can map coordinates with a pair of
values (x, y)
The slope is the average rate of
change between 2 points (calculated
as change in y divided by change in
x)

5. An introduction!
What if we wanted to find the distance
between two points in the coordinate
plane? What would the distance be
between, let’s say, (3,5) and (5,5)?
Since there is no change in y, we just
subtract the change in x. The distance
between these points is 2

6. An introduction!
If we had to find the distance
between (7, 4) and (7, 7), we try to
find the slope and find that there is
no change in x, so there’s no slope
Since there’s no change in x, we just
subtract the change in y and get 3 as
the distance

7. What happens when you want
to find the distance between 2
points that have a change in y
and a change in x?
This is where the distance formula comes in

8. The distance formula
and how it is
constructed and used
Distance formula
02

9. The distance formula
Let’s say we have two points: (1, 3) and
(4, 7). Let’s represent them both on the
coordinate plane

10. The formula
A B
Do this on your own paper: Slope can be represented using rise over run, so
sketch the rise and run by drawing a line up and across. Next label what the rise
and the run are

11. The distance formula
We can see that the rise is 4 and the run is 3. We could trace those and
say that the distance is 7, but there is actually a shorter distance, which
is the direct diagonal between these 2 points
The shortest distance between 2 points is always a straight line and,
in our case, it’s a diagonal. When questions ask for you to find the
distance, they most likely always mean the shortest distance
A

12. We now have a right triangle with 2 legs of 4 and 3, and a
hypotenuse that is unknown. We need to find it, and we can
use the Pythagorean theorem, which is a2 + b2 = c2 (a and b
are the legs and c is the hypotenuse)
When we square the legs and add them, we get 42 + 32 = c2
which equals 25 = c2. We get C by taking the square root of
25, which is 5. 5 is the hypotenuse and the distance
B

13. Pythagorean
theorem

14. Pythagorean theorem
We used the Pythagorean theorem, but
let’s change a few things in our formula so
that we can just plug in values
a2 was just the square of our rise, which is
change in y. Replace a with (y2 – y1). b2 is
the square of our run (change in x).
Replace b with (x2 – x1). Now we have:
We want the distance, not distance
squared, so we take the square root of both
sides, and… that’s the distance formula!
01
02
03
a² + b² = c²
(y2 – y1)² + (x2 – x1)² = c²

15. The distance
formula!
Now that you know where this formula comes from, you will never
forget it. Use the distance formula to find the shortest distance
between any two points!
d=√(y2-y1)² +(x2-x1)²

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