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Investigating the Midpoint and Length of a Line Segment

Investigating the Midpoint and Length of a Line Segment. Definition Midpoint:. The point that divides a line segment into two equal parts. . Developing the Formula for the Midpoint of a Line Segment.

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Investigating the Midpoint and Length of a Line Segment

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  1. Investigating the Midpoint and Length of a Line Segment Definition Midpoint: The point that divides a line segment into two equal parts. Developing the Formula for the Midpoint of a Line Segment

  2. 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 C A B  = -1  • MAB = (-1, 4) 6 + (-4) 2 D = 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. 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) -5 + 3 2 -4 + 2 2 = -1 = -1 H S  • MGH = (-1, -1)   1 + 6 2 2 + (-3) 2 T G = 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 xA+ xB , yA + yB 2 2 MAB =

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

  7. 2. Challenge: Given the end point of A(-2, 5) and midpoint of (4, 4), what is the other endpoint, B. -2+ xB 2 5+ yB 2 = 4 = 4 -2 + xB = 4(2) 5 + yB = 4(2) -2+ xB , 5 + yB 2 2 (4, 4) = 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) C A B 3 – (-5) = 8 units 10 units 8 units D 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. 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 dGH= 100 √ H 3 – (-5) = 8 units dGH = 10 units G 2 – (-4) = 6 units

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

  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 dGH = √(-6–0)2 +(-2+6)2 dAB= 80 dCD= 160 dGH= 52 √ √ √ dCD = 12.64 units dAB = 8.94 units dGH= 7.21 units

  14. 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 dFP = √(6–9)2 +(2-5)2 dDP= 32 dEP= 25 dFP= 18 √ √ √ dEP = 5 units dEP = 5.66 units • Store F should receive the call. dFP= 4.24 units

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

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