1.3 Distance and Midpoint Formulas

Distance and Midpoint Formulas
The student will be able to (I can):
• Find the distance between two given points.
• Find the midpoint of two given points.
• Find the coordinates of an endpoint given one endpoint
and a midpoint.
• Find the coordinates of a point a fractional distance from
one end of a segment.

Let’s look at a right triangle that is on a coordinate plane.
Recall that
2
2 2
2
2 2
( )
a b b
c a
c
+ = = +
a
b
c
2
2 2
c b
a
= +
2
2
c b
a
= +
2
2
9 16
4
3
= + = +
25 5
= =
●
●

distance formula – given two points (x1, y1) and (x2, y2), the
distance between them is given by
Example: Use the Distance Formula to find the distance
between F(3, 2) and G(-3, -1)
( ) ( )
2 2
1
2 1
2
d x y
y
x
= − + −
x1 y1 x2 y2
3 2 –3 –1
( ) ( )
2 2
3 3 1 2
FG = − − + − −
( ) ( )
2 2
6 3 36 9
= − + − = +
45 3 5
= =
Remember that the square of a negative number is
positive.

Problems 1. Find the distance between (9, –1) and
(6, 3).
A. 5
B. 25
C. 7
D. 13

Problems 1. Find the distance between (9, –1) and
(6, 3).
A. 5
B. 25
C. 7
D. 13
( ) ( )
( )
2
2
6 9 3 1
d = − + − −
( )
2 2
3 4 25 5
= − + = =

Problems 2. Point R is at (10, 15) and point S is at (6,
20). What is the distance RS?
A. 1
B.
C. 41
D. 6.5
41

Problems 2. Point R is at (10, 15) and point S is at (6,
20). What is the distance RS?
A. 1
B.
C. 41
D. 6.5
41
( ) ( )
2 2
6 10 20 15
d = − + −
( )
2 2
4 5 41
= − + =

The coordinates of a midpoint are the averages of the
coordinates of the endpoints of the segment.
1 3 2
1
2 2
− +
= =
C A T
(5, 6)
D
O
G
x-coordinate:
y-coordinate:
2 8 10
5
2 2
+
= =
4 8 12
6
2 2
+
= =

midpoint formula – the midpoint M of with endpoints
A(x1, y1) and B(x2, y2) is found by
AB
1 1
2 2
,
2
y
2
x
M
x y
+ +
 
 
 
0
A
B
x1 x2
y1
y2
●
M
average of
x1 and x2
average of
y1 and y2

Example Find the midpoint of QR for Q(–3, 6) and
R(7, –4)
x1 y1 x2 y2
Q(–3, 6) R(7, –4)
2
1 3 7 4
2
2 2 2
x
x + +
=
−
= =
2
1 2
1
2 2 2
6 4
y
y + +
=
−
= =
M(2, 1)

Problems 1. What is the midpoint of the segment
joining (8, 3) and (2, 7)?
A. (10, 10)
B. (5, –2)
C. (5, 5)
D. (4, 1.5)

joining (8, 3) and (2, 7)?
A. (10, 10)
B. (5, –2)
C. (5, 5)
D. (4, 1.5)
8 2 10
5
2 2
+
= =
3 7 10
5
2 2
+
= =

joining (–4, 2) and (6, –8)?
A. (–5, 5)
B. (1, –3)
C. (2, –6)
D. (–1, 3)

joining (–4, 2) and (6, –8)?
A. (–5, 5)
B. (1, –3)
C. (2, –6)
D. (–1, 3)
4 6 2
1
2 2
− +
= =

Sidebar:
If you are given an endpoint and a midpoint, you will then
need to find the other endpoint. While you can use the
midpoint formula and Algebra to find the missing
coordinates, I find it much easier to take advantage of the
definition – the distance between each should be the same.
Example: If one endpoint is at (1, 7) and the midpoint is at
(6, 3), what are the coordinates of the other endpoint?
(11, –1)
1 7
5 4
6 3
 
+ −
 
 
6 3
5 4
11 -1
 
+ −
 
 

Problem 3. Point M(7, –1) is the midpoint of ,
where A is at (14, 4). Find the
coordinates of point B.
A. (7, 2)
B. (–14, –4)
C. (0, –6)
D. (10.5, 1.5)
AB

Problem 3. Point M(7, –1) is the midpoint of ,
where A is at (14, 4). Find the
coordinates of point B.
A. (7, 2)
B. (–14, –4)
C. (0, –6)
D. (10.5, 1.5)
AB
14 4
7 5
7 1
 
− −
 
−
 
7 1
7 5
0 6
−
 
− −
 
 
−

• To find the point that is a fractional distance from an
endpoint, find the distance for each axis and multiply that
distance by the fraction.
• Then, add that amount to the first coordinate.
• Example: Segment has coordinates A(–7,2) and F(5,8).
Find the coordinates of X that is ⅙ from A to F.
The distance between the x-coordinates is 5 – (–7) = 12
The distance between the y-coordinates is 8 – 2 = 6
X(–5, 3)
AF
( ) ( )
1 1
12 2, 6 1
6 6
= =

1.3 Distance and Midpoint Formulas

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1.3 Distance and Midpoint Formulas