Obj. 7 Midpoint and Distance Formulas

Obj. 7 Midpoint and Distance
Objectives:
The student is 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.
C T

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

-2 2 4 6 8 10
-2
2
4
6
8
10
x
y
D
G

-2 2 4 6 8 10
-2
2
4
6
8
10
x
y
x-coordinate:
2 8 10
5
2 2
+
= =
D
G

-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
+
= =
D
G

-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

midpoint
formula
The midpoint M of with endpoints
A(x1, y1) and B(x2, y2) is found by
AB
0
A
B
x1 x2
y1
y2

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

Example Find the midpoint of QR for Q(—3, 6) and
R(7, —4)

R(7, —4)
x1 y1 x2 y2
Q(—3, 6) R(7, —4)

R(7, —4)
x1 y1 x2 y2
Q(—3, 6) R(7, —4)
21x 3x 7 4
2
2 2 2
+ +
= = =
−

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+ +
=
−
= =

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)

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)

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

joining (—4, 2) and (6, —8)?
A. (—5, 5)
B. (1, —3)
C. (2, —6)
D. (—1, 3)

joining (—4, 2) and (6, —8)?
A. (—5, 5)
B. (1, —3)
C. (2, —6)
D. (—1, 3)
− +
= =
4 6 2
1
2 2
( )+ − −
= = −
2 8 6
3
2 2

If we are given an endpoint and the
midpoint, we can use the distances
between them to locate the missing
endpoint.
Example: T is the midpoint of SP. S has
coordinates (5, —7) and T is at (2, 5). Find
the coordinates of P.

endpoint.
midpoint — endpoint = distance
x-coordinates: 2 — 5 = —3

endpoint.
y-coordinates: 5 — (—7) = 12

endpoint.
new endpoint = midpoint + distance
P(2 + —3, 5 + 12)

endpoint.
new endpoint = midpoint + distance
P(2 + —3, 5 + 12) = P(—1, 17)

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

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
− = −7 14 7 − − = −1 4 5
( ) ( )+ − − + − = −B 7 ( 7), 1 ( 5) B 0, 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 2222 2
or c ( acb ba )
y
x
a
b
c
= +2 22
c a b
= + 22
c a b
22
164 93= + = +
25 5= =
●
●

Length of a =
Length of b =
y
x
a
b
c
−2 1x x
−2 1y y
●
●

Length of a =
Length of b =
so
y
x
a
b
c
−2 1x x
−2 1y y
( ) ( )
2 22
2 1 2 1c x x y y= − + −
●
●

Length of a =
Length of b =
so
or
y
x
a
b
c
−2 1x x
−2 1y y
( ) ( )
2 22
2 1 2 1c x x y y= − + −
( ) ( )= − + −
2 2
2 1 2 1c x x y y
●
●

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= − + −

distance
formula
( ) ( )
2
1
2
2 2 1d xx y y= − + −
x1 y1 x2 y2
3 2 —3 —1
( ) ( )= + −− −−
2 2
F 33 1G 2

distance
formula
( ) ( )
2
1
2
2 2 1d xx y y= − + −
x1 y1 x2 y2
3 2 —3 —1
( ) ( )= + −− −−
2 2
F 33 1G 2
( ) ( )2 2
6 3 36 9= − + − = +
= = ≈45 3 5 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

(6, 3).
A. 5
B. 25
C. 7
D. 13
( ) ( )= − + − −
22
d 6 9 3 ( 1)

(6, 3).
A. 5
B. 25
C. 7
D. 13
( ) ( )
( ) ( )
= − + − −
= − +
=
22
2 2
d 6 9 3 ( 1)
3 4
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

A. 1
B.
C. 41
D. 6.5
41
( ) ( )= − + −
2 2
d 6 10 20 15

A. 1
B.
C. 41
D. 6.5
41
( ) ( )
( )
= − + −
= − + = +
=
2 2
2 2
d 6 10 20 15
4 5 16 25
41

Obj. 7 Midpoint and Distance Formulas

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Obj. 7 Midpoint and Distance Formulas