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.

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)

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

 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)

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 2 2

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 = -2 + 6 , -1 + 3 2 2 MCD = 7 + (-5) , 1 + (-3) 2 2 MAB = (2, 1) MCD = (1, -1) MGH = 0 + 9 , -6 + (-2) 2 2 MGH = (9/2, -4)

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 , 5 + yB 2 2 = 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)

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

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

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

S  T dST2 = 52 + 52 dST2 = 50 dST= 50 √ 2 – (-3) dST = 7.07 units b) S(1, 2) and T(6, -3) dST2 = 52 + 52 dST2 = 50 T S  dST= 50 √ 2 – (-3) = 5 units dST = 7.07 units 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) 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 = 12.64 units dGH = √(-6–0)2 +(-2+6)2 dGH= 52 √ dGH= 7.21 units

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 = 32 √ 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

c) Find a point that would be the same distance from two of these stores. MDF = 2 + 9 , -2 + 5 2 2 MDE = 2 + 9 , -2 – 2 2 2 MDE = (11/2, -2) MDF = (11/2, 3/2) MEF = 9 + 9 , -2 + 5 2 2 MEF = (9, 3/2)