1.1.5 midpoint and partition formulas
TRANSCRIPT
-
Midpoint and Partition 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.and a midpoint.
Find the coordinates of a point a fractional distance from one end of a segment.
-
The coordinates of a midpoint are the averages of the coordinates of the endpoints of the segment.
C A T
1 3 21
2 2
+= =
-
-2 2 4 6 8 10
2
4
6
8
10
x
y
(5, 6)D
O
G
-2
x-coordinate:
y-coordinate:
2 8 105
2 2
+= =
4 8 126
2 2
+= =
-
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+ +
A
B
y
y2
Maverage of y1 and y2
0
A
x1 x2
y1
average of x1 and x2
-
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 1yy 6 4+ +
=
= =212
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
D. (10.5, 1.5)
14 7 7 = 7 7 0 =
( )4 1 5 = 1 5 6 =
-
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
A4 8 11 1
C , 2 2
+
( )C 2,5
E
-
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
A4 8 11 1
C , 2 2
+
( )C 2,5C 4 2 11 5 + +
E
C 4 2 11 5B ,
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
A4 8 11 1
C , 2 2
+
( )C 2,5C 4 2 11 5 + +
B
E
C 4 2 11 5B ,
2 2
+ +
( )B 1,8
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
A4 8 11 1
C , 2 2
+
( )C 2,5C 4 2 11 5 + +
B
E
C 4 2 11 5B ,
2 2
+ +
( )B 1,8
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) AB
1 2 1 2ax bx ay by, a b a b
+ + + +
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
( ) ( ) ( ) ( )2 3 1 15 2 16 1 4P ,
1 2 1 2
+ + + +
( )P 3, 12