1 . This question aims at investigating the design of a square farm by Voronoi diagrams and graph theory .
The diagram below shows the Voronoi diagram of a square farm OABCDE bounded by the coordinate axes , the lines x � 30 and y � 30 , where 1 unit represents 1m . y
30
A B C
D
E O 30 x
The owner of the farm wishes to create two roads OB and OD to let his car to travel in the farm . He first drafts the directions of the roads such that
AOB ˆ BOD ˆ DOE ˆ � � � � rad . It is given that ABC and CDE are straight lines .
6
Let area of OAB : area of OBCD : area of ODE � 1 : r : 1.
( a ) ( i ) Find DE .
( ii ) Show that the area of the triangle ODE is
2
260 m .
( iii ) Hence , write down r .
Consider the case when r � 1.
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