1) The document discusses the distance and midpoint formulas. The distance formula calculates the distance between two points on a coordinate plane using their x and y coordinates, while the midpoint formula calculates the midpoint between two points. 2) An example uses the distance formula to calculate the distance between points (2, -6) and (-3, 6), finding it to be 13. 3) Another example finds the midpoint between points (1, 2) and (-5, 6) to be (-2, 8) using the midpoint formula.

1. The Distance and
Midpoint Formulas!
To be used when you want to find the
distance between two points or the
midpoint between two points

2. You have learned…
1
The Pythagorean
a c Theorem
a2 + b2 = c 2
b 2
When we are finding c we are really finding
the distance between angles 1 and 2
If we solve for c we get c = √(a2 + b2)
This is how the distance formula is derived –
it is useful when we know coordinates

3. The Distance Formula!
The distance formula is
d = √((y2 – y1)2 + (x2 – x1)2)
And is used to find the distance between two
points on the coordinate plane
Let’s practice one!

4. Example:
What is the distance between (2, -6) and (-3, 6)?
First, identify x1, y1, x2 and y2
x1 = 2 y1 = -6
x2 = -3 y2 = 6
Now, we use the formula:
d = √((6 – -6)2 + (-3 – 2)2) = √(122 + (-52))
d = 13

5. The Midpoint Formula!
The midpoint formula is used to find the
coordinate that is the exact midpoint between
two other coordinates
The x-coordinate of the midpoint is found by
(x2 + x1)/2
The y-coordinate of the midpoint is found by
(y2 + y1)/2
So, the coordinate of the midpoint is:
(x2 + x1)/2, (y2 + y1)/2

6. Example:
What is the coordinate of the midpoint between
(1, 2) and (-5, 6)?
Again, identify x1, y1, x2 and y2
x1 = 1 y1 = 2
x2 = -5 y2 = 6
Now use the formula:
x-coordinate: (1 + -5)/2 = -2
y-coordinate: (2 + 6)/2 = 8
So, the midpoint is located at the coordinate (-2, 8)

7. Another Example
On the coordinate plane, it is given that the
midpoint of points A and B is (5, 7). If point
A is located at (-1, 2), where is point B
located?
In this case, we know the midpoint and the
coordinate of point A. In a sense, we need
to work backwards. Let’s define what we
have:

8. x1 = -1 y1 = 2
x2 = ? y2 = ?
So we know…
(-1 + x2)/2 = 5 (2 + y2)/2 = 7
-1 + x2 = 10 (2 + y2) = 14
x2 = 11 y2 = 12
So the coordinate of point B is (11, 12)