Obj. 5 Midpoint and Distance Formulas

Obj. 5 Midpoint and Distance
The student will be able to (I can):
• Find the midpoint of two given points.
• Find the coordinates of an endpoint given one endpoint
aanndd aa mmiiddppooiinntt..
• Find the distance between two points.

The coordinates of a midpoint are the
averages of the coordinates of the
endpoints of the segment.
C A T
− +
1 3 2
1
= =
2 2

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

midpoint
formula
The midpoint M of AB
with endpoints
A(x1, y1) and B(x2, y2) is found by
 x + x y + y 
 
 
1 2 1 2 M ,
2 2
A
B
y2
y
M
●
average of
y1 and y2
0
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 y2
Q(—3, 6) R(7, —4)
+ −
+
1 2 x x 3 7 4
2
= = =
2 2 2
1 2 2
1
y + y
6
+ 4
=
−
= =
2 2
2
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 10
+
= = 3 7 10
5
2 2
5
= =
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 2
1
= =
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 2 2 2 or a + b = c (c = a +b )
y
x
a
b
c
●
●
2 2 2 c = a +b
2 2 c = a +b
2 2 = 3 + 4 = 9 + 16
= 25 = 5

distance
formula
Given two points (x1, y1) and (x2, y2), the
distance between them is given by
( ) ( ) 2
d = x2 − x 1
+ y2 − y1
2
Example: Use the Distance Formula to find
the distance between F(3, 2) and G(-3, -1)
x1 y1 x2 y2
33 22 ——33 ——11
( )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 ppppoooossssiiiittttiiiivvvveeee.

Problems 1. Find the distance between (9, —1) and
(6, 3).
A. 5
B. 25
C. 7
D. 13
( ) ( ( ))2 2 d = 6 − 9 + 3 − −1
( )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.
41
C. 41
D. 6.5
( )2 ( )2 d = 6 − 10 + 20 − 15
( )2 2 = −4 + 5 = 41

Obj. 5 Midpoint and Distance Formulas

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Obj. 5 Midpoint and Distance Formulas