Investigating the Midpoint and Length of a Line Segment

Developing the Formula for the Midpoint of a Line Segment

DefinitionMidpoint: 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)

b) C(1, 6) and D(1, -4)

A B

D

C

-5 + 3 2= -1\MAB = (-1, 4)

6 + (-4) 2= 1\MCD = (1, 1)

B. Describe how you found the midpoint of each line segment.

• To find the midpoint of AB, add the x-coordinates together and divide by 2

• To find the midpoint of CD, add the y-coordinates together and divide by 2

C. 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 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)

b) S(1, 2) and T(6, -3)

G

H

T

S

-4 + 2 2

= -1\MGH = (-1, -1)

1 + 6 2

= 7/2\MST = (7/2, -1/2)

-5 + 3 2

= -1

2 + (-3) 2= -1/2

D. In your group, 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

2 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)

c) G(0, -6) and H(9, -2)

MAB = -2 + 6 , -1 + 3 2 2

MCD = 7 + (-5) , 1 + (-3) 2 2

MGH = 0 + 9 , -6 + (-2) 2 2

MAB = (2, 1) MCD = (1, -1)

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 , 5 + yB

2 2

= 4-2 + xB

2

= 4 5 + yB

2 -2 + xB = 4(2) xB = 8 + 2

xB = 10

5 + yB = 4(2)

yB = 8 - 5

yB = 3

\The other end point is B (10, 3)

3. Super Challenge: If you have the endpoints from question 1a, what is the equation of the line that goes through the original two points, and what is the equation of the perpendicular line that goes through the midpoint?

mAB = 3 – (-1)6 – (-2)

mAB = 1/2

-1 = ½(-2) + b-1 = -1 + bb = 0

\ y = ½x

A

B

3. Super Challenge: If you have the endpoints from question 1a, what is the equation of the line that goes through the original two points, and what is the equation of the perpendicular line that goes through the midpoint?

\ y = -2x + 5

m = -2

1 = -2(2) + b

1 = -4 + bb = 5

A

B

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)

A B

D

C 3 – (-5) = 8 units

6 – (-4) = 10 units

Developing the Formula for the Length of a Line Segment

8 units 10 units

B. Describe how you found the length of each line segment.

• To find the length of AB, subtract the x-coordinates

• To find the length of CD, subtract the y-coordinates

C. 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)

G

H

dGH2 = 62 + 82

dGH = 10 units

dGH= 100√

dGH2 = 100

2 – (-4) = 6 units

3 – (-5) = 8 units

b) S(1, 2) and T(6, -3)

T

S

dST2 = 52 + 52

dST = 7.07 units

dST= 50√

dST2 = 50

6 – 1 = 5 units

2 – (-3) = 5 units

D. In your group, 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)

c) G(0, -6) and H(9, -2)

dAB = √(6+2)2 +(3+1)2

dAB= 80√

dAB = 8.94 units

dCD = √(-5–7)2 + (-3–1)2

dCD= 160√

dCD = 12.64 units

dGH = √(9–0)2 +(-2+6)2 dGH= 97√dGH= 9.84 units

2. Challenge: If a cell phone is located at point P(-2, 7) and the nearest telecommunication towers are at points A(0, 0), B(3, 3), and C(4, -1). To which tower should the call go?

dAP = √(-2-0)2 +(7-0)2

dAP = 53√

dAP = 7.23 units

dBP = √(-2–3)2 + (7–3)2

dBP= 41√

dBP = 6.4 units

dCP = √(-2–4)2 +(7+1)2 dCP= 100√dCP= 10 units

\ Tower B should receive the call.

3. Super 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 = 5.66 units

dEP = √(6–9)2 + (2+2)2

dEP= 25√

dEP = 5 units

dFP = √(6–9)2 +(2-5)2 dFP= 18√dFP= 4.24 units

\ Store F should receive the call.

b) Would your answer in part i) always be the best for a pizza company? Can you think of circumstances that would change your response in i?• If store F was too busy, you may

choose the next closest store

c) Find a point that would be the same distance from two of these stores.

MDE = 2 + 9 , -2 – 2 2 2

MDF = 2 + 9 , -2 + 5 2 2

MEF = 9 + 9 , -2 + 5 2 2

MDE = (11/2, -2) MDF = (11/2, 3/2)

MEF = (9, 3/2)