-
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.
-
The coordinates of a midpoint are the averages of the
coordinates of the endpoints of the segment.
1 3 21
2 2
+= =
C A T
-
-2 2 4 6 8 10
-2
2
4
6
8
10
x
y
x-coordinate:
y-coordinate:
2 8 105
2 2
+= =
4 8 126
2 2
+= =
(5, 6)D
O
G
-
midpoint formula
The midpoint M of with endpoints A(x1, y1) and B(x2, y2) is
found by
AB
1 12 2M , 2 2
yxx 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 y2Q(3, 6) R(7, 4)
21x 3x 7 4 22 2 2
+ += = =
21 2 1y
2 2
y 6
2
4+ +=
= =
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)
8 2 105
2 2
+= =
3 7 105
2 2
+= =
-
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 21
2 2
+= =
-
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 =
-
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 2or b c b(ca a )+ = = +y
x
a
bc
22 2c ba= +22c ba= +22 164 93= + = +
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
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.
-
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 23 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
( ) ( )2 2
d 6 10 20 15= +
( )2 24 5 41= + =
-
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 1C ,
2 2
+
( )C 2,5
-
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 1C ,
2 2
+
( )C 2,5C 4 2 11 5
B , 2 2
+ +
( )B 1,8
-
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 1C ,
2 2
+
( )C 2,5C 4 2 11 5
B , 2 2
+ +
( )B 1,8
B
2 8 5 1D ,
2 2
+
( )D 5,2
-
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 1C ,
2 2
+
( )C 2,5C 4 2 11 5
B , 2 2
+ +
( )B 1,8
B
2 8 5 1D ,
2 2
+
( )D 5,2
D
-
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 4P ,
1 2 1 2
+ + + +
( )P 3, 12
