### Midpoint and Length of Line Segmentx

```Investigating the Midpoint and
Length of a Line Segment
Developing the Formula for the Midpoint of
a Line Segment
Definition
Midpoint: The point that divides a line
segment into two equal parts.
A. Graph the following pairs of points on graph paper.
Connect points to form a line segment.
Investigate ways to find the midpoint of the segment.
Write the midpoint as an ordered pair.
a) A(-5, 4) and B(3, 4)
MAB = (-1, 4)
A
C
B


-5 + 3
2
= -1
b) C(1, 6) and D(1, -4)
6 + (-4)
2
=1
MCD = (1, 1)
D
Describe how you found the midpoint of each
line segment.
• To find midpoint of AB, add x-coordinates
together and divide by 2
• To find midpoint of CD, add y-coordinates
together and divide by 2
B. Graph the following pairs of points on graph paper. Connect
points to form a line segment. Find the midpoint using your
procedure described in part A. If your procedure does not
work, see if you can discover another procedure that will
work.
a) G(-4, -5) and H(2, 3)
S

b) S(1, 2) and T(6, -3)
1+6
2
= 7/2
2 + (-3)
2
= -1/2
MST = (7/2, -1/2)
G
H

-5 + 3
-4 + 2
2
2
= -1
= -1
MGH = (-1, -1)
T
C. Compare your procedures and develop a
formula that will work for all line segments.
Line segment with end points, A(xA, yA)
and B(xB, yB), then the midpoint is
MAB =
xA + xB , y A + yB
2
2
D. Use the formula your group created in part C to
solve the following questions.
1. Find the midpoint of the following pairs of points:
a) A(-2, -1) and B(6, 3) b) C(7, 1) and D(-5, -3)
MAB =
-2 + 6 , -1 + 3
2
2
MCD =
7 + (-5) , 1 + (-3)
2
2
MCD = (1, -1)
MAB = (2, 1)
c) G(0, -6) and H(9, -2)
MGH =
0 + 9 , -6 + (-2)
2
2
MGH = (9/2, -4)
2. Challenge: Given the end point of A(-2, 5) and
midpoint of (4, 4), what is the other endpoint, B.
(4, 4) =
-2 + xB
-2 + xB = 4
2
-2 + xB = 4(2)
xB = 8 + 2
xB = 10
2
, 5 + yB
2
5 + yB = 4
2
5 + yB = 4(2)
yB = 8 - 5
yB = 3
The other end point is B (10, 3)
Developing the Formula for the Length of a
Line Segment
A. Graph the following pairs of points on graph paper. Connect
points to form a line segment.
Investigate ways to find the length of the each segment.
a) A(-5, 4) and B(3, 4)
3 – (-5) = 8 units
b) C(1, 6) and D(1, -4)
6 – (-4) = 10 units
A
C
B
8 units
D
10 units
Describe how you found the length of each
line segment.
• To find length of AB, subtract the xcoordinates
• To find length of CD, subtract the ycoordinates
B. Graph the following pairs of points on graph paper.
Connect points to form a line segment. Find the length
using your procedure described in part B. If your
procedure does not work, see if you can discover another
procedure that will work.
a) G(-4, -5) and H(2, 3)
dGH2 = 62 + 82
dGH2 = 100
 H
dGH= √100
3 – (-5)
= 8 units
dGH = 10 units
G
2 – (-4)
= 6 units
b) S(1, 2) and T(6, -3)
dST2 = 52 + 52
dST2 = 50
S
dST= √50
dST = 7.07 units
2 – (-3)
= 5 units
T
6–1
= 5 units
C. Compare your procedures and develop a
formula that will work for all line segments.
Line segment with end points, A(xA, yA)
and B(xB, yB), then the length is
dAB2 = (xB – xA)2 + (yB – yA)2
dAB = √(xB – xA)2 + (yB – yA)2
E. Use the formula your group created in part D to
solve the following questions.
1. Find the midpoint of the following pairs of points:
a) A(-2, -1) and B(6, 3) b) C(7, 1) and D(-5, -3)
dAB = √(6+2)2 +(3+1)2
dAB= √80
dCD = √(-5–7)2 + (-3–1)2
dCD= √160
dAB = 8.94 units
dCD = 12.64 units
c) G(0, -6) and H(9, -2)
dGH = √(-6–0)2 +(-2+6)2
dGH= √ 52
dGH= 7.21 units
2. Challenge: A pizza chain guarantees delivery in 30
minutes or less. The chain therefore wants to minimize
the delivery distance for its drivers.
a) Which store should be called if a pizza is to be
delivered to point P(6, 2) and the stores are located at
points D(2, -2), E(9, -2), F(9, 5)?
dDP = √(6-2)2 +(2+2)2
dDP = √ 32
dEP = √(6–9)2 + (2+2)2
dEP= √25
dEP = 5.66 units
dEP = 5 units
dFP = √(6–9)2 +(2-5)2
dFP= √18
dFP= 4.24 units
 Store F should
c) Find a point that would be the same distance
from two of these stores.
MDE =
2 + 9 , -2 – 2
2
2
MDE = (11/2, -2)
MEF =
MDF =
2 + 9 , -2 + 5
2
2
MDF = (11/2, 3/2)
9 + 9 , -2 + 5
2
2
MEF = (9, 3/2)
```