1. 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.

2. D C  B   -5 + 3 2 A  = -1 MAB = (-1, 4) 6 + (-4) 2 = 1
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)
b) C(1, 6) and D(1, -4)
-5 + 3 2 D C
A 
 B 
= -1 
MAB = (-1, 4)
6 + (-4) 2
= 1
MCD = (1, 1)

3. 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

4.  H S    T -4 + 2 -5 + 3 2 2 = -1 = -1 MGH = (-1, -1) 1 + 6
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)
b) S(1, 2) and T(6, -3)
-4 + 2 2
-5 + 3 2
= -1
= -1
G 
 H T
S 
MGH = (-1, -1)  
1 + 6 2
2 + (-3) 2
= 7/2
= -1/2
MST = (7/2, -1/2)

5. 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 , yA + yB

6. 1. Find the midpoint of the following pairs of points:
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)
c) G(0, -6) and H(9, -2)
MAB = ,
MCD =
7 + (-5) , 1 + (-3)
MAB = (2, 1)
MCD = (1, -1)
MGH =
0 + 9 , (-2)
MGH = (9/2, -4)

7. The other end point is B (10, 3)
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 , yB
= 4
-2 + xB 2
= 4
5 + yB 2
-2 + xB = 4(2)
5 + yB = 4(2)
xB = 8 + 2
yB = 8 - 5
xB = 10
yB = 3
The other end point is B (10, 3)

8. 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)
   
b) C(1, 6) and D(1, -4) D C
3 – (-5) = 8 units
A 
 B
10 units
8 units
6 – (-4) = 10 units

9. Describe how you found the length of each line segment.
To find length of AB, subtract the x-coordinates
To find length of CD, subtract the y-coordinates

10.  H dGH2 = 62 + 82 dGH2 = 100 dGH= 100 √ 3 – (-5) = 8 units
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 =
dGH2 = 100
G 
 H
dGH= 100 √
3 – (-5)
= 8 units
dGH = 10 units
2 – (-4)
= 6 units

11. S  T dST2 = 52 + 52 dST2 = 50 dST= 50 √ 2 – (-3) dST = 7.07 units
b) S(1, 2) and T(6, -3)
dST2 =
dST2 = 50 T
S 
dST= 50 √
2 – (-3)
= 5 units
dST = 7.07 units
6 – 1
= 5 units

12. 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

13. 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)
c) G(0, -6) and H(9, -2)
dAB = √(6+2)2 +(3+1)2
dCD = √(-5–7)2 + (-3–1)2
dAB= 80 √
dCD= 160 √
dAB = 8.94 units
dCD = units
dGH = √(-6–0)2 +(-2+6)2
dGH= 52 √
dGH= 7.21 units

14. Store F should receive the call.
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
dEP = √(6–9)2 + (2+2)2
dDP = √
dEP= 25 √
dEP = 5.66 units
dEP = 5 units
dFP = √(6–9)2 +(2-5)2
Store F should receive the call.
dFP= 18 √
dFP= 4.24 units

15. c) Find a point that would be the same distance from two of these stores.
MDF =
2 + 9 ,
MDE =
2 + 9 , -2 – 2
MDE = (11/2, -2)
MDF = (11/2, 3/2)
MEF =
9 + 9 ,
MEF = (9, 3/2)