3.)Buried Treasure Pi map with directions to its location. The map has simp rate

Question

3.)Buried Treasure Pi map with directions to its location. The map has simp rates have buried treasure on a deserted island, and you have found the le directions: 1) Go to tree A. 2) Go exactly half way from tree A to tree B and stop. 3) Th en go 1/3 of the way from this location to tree C. and stop. The treasure is buried 10 feet underground at this location. The problem is - they didn't label which tree is A, B, or C! (a) Suppose you have located on a grid map of the island, the three trees at (x, y) positions:

Explanation / Answer

part a:

let the assignment of tree locations be:

A=(0,30)

B=(60,30)

C=(60,-30)

half way from tree A to tree B=D=((0,30)+(60,30))/2=(30,30)

location of (x,y) =location of D +(1/3)*distance between D and C

=(30,30)+(1/3)*((60,-30)-(30,30))=(30,30)+(1/3)*(30,-60)=(40,10)

now consider the reverse order:

A=(60,-30)

B=(60,30)

C=(0,30)

half way point D=((60,-30)+(60,30))/2=(60,0)

location of (x,y)=(60,0)+(1/3)*((0,30)-(60,0))=(60,0)+(1/3)*(-60,30)=(40,10)

so in both cases we got the same value.

part b:

location of (x,y) in vector terms:

half way point of A and B=(A+B)/2

1/3 distance from this point to C=((A+B)/2)+(1/3)*(C-0.5*(A+B))=(A+B+C)/3

so any way you chose A,B and C, the final location will be same i.e. (A+B+C)/3

part c:

in case of 4 trees:

half way point of A and B=(A+B)/2

from their 1/3rd way to C is located at =0.5*(A+B)+(1/3)*(C-0.5*(A+B))=(A+B+C)/3

from their 1/4th way to D is located at=(1/3)*(A+B+C)+(1/4)*(D-(1/3)*(A+B+C))=(A+B+C+D)/4

so in case of 4 trees also, we arrive at same location irrespective of choice of assignment of trees,

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