Voronoi Diagrams-Bones of Contentions

NUMB3RS ActivityTeacher Page 2"Geoboard" ApproachMidpoint: The midpoint of AB is located athalf of the horizontal distance from A to Band half of the vertical distance fromA to B. From A, the midpoint has a verticaldistance 3 and horizontal distance 2.Slope: The slope of AB is the verticaldistance from A to B divided by thehorizontal distance from A to B: 6 = 3 .4 2Slope of Perpendicular Bisector: Thisslope is the opposite reciprocal of the2slope of AB : − .3Episode: "Bones of ContentionCoordinate ApproachCoordinates: A(0,2) and B(4,8)⎛0+ 4 2+8⎞⎜ ⎟⎝ 2 2 ⎠Midpoint: , = ( 2, 5 )Slope: 8 − 2 = 6 =34−0 4 2Slope of Perpendicular Bisector:2−3B987B (4, 8)M6654M (2, 5)3A421A (0, 2)01 2 3 4 5 6 7 8 9Student page answers: 1a. AB : slope = 3 2, midpoint = (2, 5), slope of perpendicular bisector =− 2 ; BC : slope = , midpoint = (6, 4), slope of perpendicular bisector =3 −212 ; AC : slope = − 1 4 ,midpoint = (4, 1), slope of perpendicular bisector = 4 1b and 1c. See diagram below 2. Store B;the house at (5, 5) is in the same region (zone) as Store B, so Store B is closest to this house.3. Either Store A or Store B; because this line is the perpendicular bisector of AB , a point on theline will be the same distance from Store A and Store B 4. See diagram below 5a. The ‘best’location is the Voronoi vertex – located near (4.6,3.3) 5b. See diagram below 6. This willhappen only when the four points lie on a circle. 7. Sample answer: Driving times are assumed tobe uniform and independent of traffic.1c. 4. 5.BBDNBNote: the dashedlines are not partof the Voronoidiagram.AAAeducation.ti.com/go/NUMB3RSTom Butts, UT DallasCCC© 2005 Texas Instruments Incorporated