Find the midpoint of two given points. Find the coordinates of an endpoint given one endpoint and a midpoint. Find the distance between two points.

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

2. 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

3. -2 2 4 6 8 10
-2
2
4
6
8
10
x
y
•
x-coordinate:
y-coordinate:
2 8 10
5
2 2
+
= =
4 8 12
6
2 2
+
= =
(5, 6)
D
O
G

4. midpoint
formula
The midpoint M of with endpoints
A(x1, y1) and B(x2, y2) is found by
AB
1 12 2
M ,
2 2
yxx y+ + 
 
 
0
A
B
x1 x2
y1
y2
●
M
average of
x1 and x2
average of
y1 and y2

5. Example Find the midpoint of QR for Q(—3, 6) and
R(7, —4)
x1 y1 x2 y2
Q(—3, 6) R(7, —4)
21x 3x 7 4
2
2 2 2
+ +
= = =
−
21 2
1
y
2 2
y 6
2
4+ +
=
−
= =
M(2, 1)

6. 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)
8 2 10
5
2 2
+
= =
3 7 10
5
2 2
+
= =

7. Problems 2. What is the midpoint of the segment
joining (—4, 2) and (6, —8)?
A. (—5, 5)
B. (1, —3)
C. (2, —6)
D. (—1, 3)
4 6 2
1
2 2
− +
= =

8. 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 7 7− = 7 7 0− =
( )4 1 5− − = 1 5 6− − = −

9. Pythagorean
Theorem
In a right triangle, the sum of the squares
of the lengths of the legs is equal to the
square of the length of the hypotenuse.
2 2 2 22 2
or b c b(ca a )+ = = +
y
x
a
b
c
22 2
c ba= +
22
c ba= +
22
164 93= + = +
25 5= =
●
●

10. 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
1
2
2 2 1d xx y y= − + −
x1 y1 x2 y2
3 2 —3 —1
( ) ( )2 2
FG 3 3 1 2= − − + − −
( ) ( )2 2
6 3 36 9= − + − = +
45 6.7= ≈
Note: Remember that the square of a
negative number is positivepositivepositivepositive.

11. Problems 1. Find the distance between (9, —1) and
(6, 3).
A. 5
B. 25
C. 7
D. 13
( ) ( )( )
22
d 6 9 3 1= − + − −
( )2 2
3 4 25 5= − + = =

12. 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
d 6 10 20 15= − + −
( )2 2
4 5 41= − + =

13. Use the midpoint formula multiple times to
find the coordinates of the points that
divide into four congruent segments.
(Find points B, C, and D.)
AE
A
E
4 8 11 1
C ,
2 2
 − + −    
( )C 2,5

14. Use the midpoint formula multiple times to
find the coordinates of the points that
divide into four congruent segments.
(Find points B, C, and D.)
AE
A
E
4 8 11 1
C ,
2 2
 − + −    
( )C 2,5
C
4 2 11 5
B ,
2 2
 − + +    
( )B 1,8−

15. Use the midpoint formula multiple times to
find the coordinates of the points that
divide into four congruent segments.
(Find points B, C, and D.)
AE
A
E
4 8 11 1
C ,
2 2
 − + −    
( )C 2,5
C
4 2 11 5
B ,
2 2
 − + +    
( )B 1,8−
B
2 8 5 1
D ,
2 2
 + −    
( )D 5,2

16. Use the midpoint formula multiple times to
find the coordinates of the points that
divide into four congruent segments.
(Find points B, C, and D.)
AE
A
E
4 8 11 1
C ,
2 2
 − + −    
( )C 2,5
C
4 2 11 5
B ,
2 2
 − + +    
( )B 1,8−
B
2 8 5 1
D ,
2 2
 + −    
( )D 5,2
D

17. partitioning a
segment
Dividing a segment into two pieces whose
lengths fit a given ratio.
For a line segment with endpoints (x1, y1)
and (x2, y2), to partition in the ratio b: a,
Example: has endpoints A(—3, —16)
and B(15, —4). Find the
coordinates of P that partition
the segment in the ratio 1 : 2.
AB
1 2 1 2ax bx ay by
,
a b a b
 + +    + +
( ) ( ) ( ) ( )2 3 1 15 2 16 1 4
P ,
1 2 1 2
 − + − + −    + + 
( )P 3, 12−