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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.

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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 A BB DD CC = -1 MM AB = (-1, 4) 6 + (-4) 2 = 1 MM CD = (1, 1)

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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

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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) G G HH TT S = -1 MM GH = (-1, -1) = 7/2 MM ST = (7/2, -1/2) = (-3) 2 = -1/2

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C. Compare your procedures and develop a formula that will work for all line segments. Line segment with end points, A(x A, y A ) and B(x B, y B ), then the midpoint is M AB = x A + x B, y A + y B 2 2

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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) M AB = , M CD = 7 + (-5), 1 + (-3) 2 2 M GH = 0 + 9, -6 + (-2) 2 2 M AB = (2, 1) M CD = (1, -1) M GH = (9/2, -4)

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2.Challenge: Given the end point of A(-2, 5) and midpoint of (4, 4), what is the other endpoint, B. (4, 4) = x B, 5 + y B 2 2 = x B 2 = y B x B = 4(2) x B = x B = y B = 4(2) y B = y B = 3 TT he other end point is B (10, 3)

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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 A BB DD CC 3 – (-5) = 8 units 6 – (-4) = 10 units Developing the Formula for the Length of a Line Segment 8 units 10 units

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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

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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) G G HH d GH 2 = d GH = 10 units d GH = 100 √ d GH 2 = – (-4) = 6 units 3 – (-5) = 8 units

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b) S(1, 2) and T(6, -3) TT S d ST 2 = d ST = 7.07 units d ST = 50 √ d ST 2 = 50 6 – 1 = 5 units 2 – (-3) = 5 units

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C. Compare your procedures and develop a formula that will work for all line segments. Line segment with end points, A(x A, y A ) and B(x B, y B ), then the length is d AB 2 = (x B – x A ) 2 + (y B – y A ) 2 d AB = √(x B – x A ) 2 + (y B – y A ) 2

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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) d AB = √(6+2) 2 +(3+1) 2 d AB = 80 √ d AB = 8.94 units d CD = √(-5–7) 2 + (-3–1) 2 d CD = 160 √ d CD = units d GH = √(-6–0) 2 +(-2+6) 2 d GH = 52 √ d GH = 7.21 units

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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)? d DP = √(6-2) 2 +(2+2) 2 d DP = 32 √ d EP = 5.66 units d EP = √(6–9) 2 + (2+2) 2 d EP = 25 √ d EP = 5 units d FP = √(6–9) 2 +(2-5) 2 d FP = 18 √ d FP = 4.24 units S tore F should receive the call.

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c) Find a point that would be the same distance from two of these stores. M DE = 2 + 9, -2 – M DF = 2 + 9, M EF = 9 + 9, M DE = (11/2, -2) M DF = (11/2, 3/2) M EF = (9, 3/2)

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