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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
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