Solving Voronoi Diagram Problems: Coordinates, Bisectors, and Restaurant Choices

the diagram below is a part of warid diagram A and B are sides with the B having the coordinates of for 4 six now

is an edge Edge means a perpendicular BCT line equivalent of this perpendicular bis

sector of the line segment from A to B is yal 2x + 9 let's connect AB

perpendicular bis sector means L is perpendicular to line AB intersection let's label as M because

this is Middle Point of segment a find the coordinates of for the point a according

to the given information we know the equ for line L Y = -2x +

9 so the slope of for l equals

-2 which means the slope of ab equals

opposite reciprocal 1 over two second step since the we figured out the slope

of ab is one

half we will use y - y1 = slope * x - X1 to get the equation for line

AB we already got the slope of line AB equal 12 coordinat for B is 46

use these two information we can get linear equation of for line AB this is X1

y1 so y - 6 = 12 * x - 4 we need to figure out the coordinate of for M because this m is the intersection of

all two lines y - 6 = 12 x - 4 Y = -2x +

9 by G DC by calculator we can get coordinat of for M

SC PA menu

algebra 7 2 okay then you need to put those two

equations then enter we got coordinates for m 2 five bis sector means m is the the

midpoint of the segment a formula for midpoint to figure out the coordinates for a coordinates for a is X1

y1 mid point 25 B 4 6 let's label this is the X2 Y 2 you use

this Formula 2 = X1 + 4 over 2 5 = y1 + 6 / 2 I'll show you solve

this by hand also by graphing calculator 2 over one cross multiply X1 + 4 = 2 * 2 is 4 X1 =

0 5/ 1 cross multiply 10 = y1 + 6 cross the river change the sign 10 - 6 = y1 so y1 = 4 the coordinat

for a is 0 4 2

equals control divide put the variable as a X

plus 4 divided by two then control

menu two math actions enter enter X =

zero 5 equals control divide you can use the variable y plus 6 over

two then control menu two enter enter y =

four so we got same answer six restaurant locations labeled a b c d e and f are shown together with their

verid diagram all distance are measured in kilometers Elena wants to eat at the

closest restaurant write down the restaurant she should go to if she is at

27 27 you could have say you could have say this location has

the same cell with the restaurant B answer is B why you have to use the property of oid diagram cell to

answer this question any point in this cell is a closer to restaurant B than any other

restaurant A2 01 when restaurant a is closed 01 is here since Resturant a is

close closed then choose the closest restaurant to this location I need to pay attention to the

Mark only one mark means you need to give the answer directly no work needed from the diagram you could say it's

closest to restaurant F answer is f that's all normally no work needed for one

Mark a little bit of work for two marks plenty of work needed for three marks or above restaurant C is at 78 and

restaurant d is at 75 let's label the coordinates equation of perpendicular

bis sector of CD let's connect CD first perpendicular to this

Edge this is the perpendicular bis seor line figure out this

midpoint 7 + 7 / 2 8 + 5 over 2 so we get coordinates for this midpoint is 7

6.5 so y = 6.5 is the equivalent of perpendicular bis sector of

CD Resturant B is at 36 find the equalent of perpendicular bis sector of BC five marks so you really need to show

the work step by step let's connect this BC write down the coordinates for B and C

first let's label the mid point for Segment BC and label X1 y1 X2 Y2 according to this formula

midpoint of for Segment BC it's 3 + 7 / 2 6 + 8 / 2 equals work this

out 10 / 2 is 5 14 / 2 is is seven second step we need to use this

formula to figure out the slope of line BC

equal Y 2 - [Music] y1 X2 -

X1 = 2 over 4 simplify 1/ 2 bis sector of BC that is the midpoint n perpendicular means we need to figure

out slope of perpendicular line to

BC write down the symbol like this slope of a perpendicular line of BC

equals opposite reciprocal 2 since we are looking for the equ of for perpendicular bis

sector perpendicular bis sector will pass through

the midd point and we will use this equalent of a straight line to figure out the equalent of a

perpendicular B vctor of BC let's label X1 y1 y - 7 = -2 * x - 5 since no

specific form of a linear function required this will be the solution to question

C hence find the coordinates of the point which is of equal distance from BC D we do know

vertex is a for equal distance from BC D so this is the point we are looking for let's label as the

p p is the intersection of two edges these two edges we already figured out the

equation Y = 6.5 and Y - 7 = -2X - 5 we will set up for system of four

equations y - 7 = -2 * x - 5 y = 6.5 by G

DC new three seven

enter enter type those two equ separately then enter 5.2 5

6.5 is the coordinates of this vertex D2 the distance of for this point

from D we will use the distance equals X1 - X2 2 + y1 - Y 2 s d = square root

5.25 - 7 s + 6.5 - 5 S control

x² parentheses 5.25 - 7

squar plus pareses 6.

5 minus 5 Square enter 2.30 as a 36 fix f a manager believes that the larger the area of the Vero

Cale the more people will come to the restaurant in that Cale State one criticism for this inter ration answer

varies for example it depends on the density of a people in the cell or the larger area doesn't necessarily result

in more people