SOLUTION: i am having trouble on finding the point equidistant for a triangle. the problem comes from a math book but i am not sure how i need to start it or work it. PROBLEM: A softba

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Question 360558: i am having trouble on finding the point equidistant for a triangle.
the problem comes from a math book but i am not sure how i need to start it or work it.
PROBLEM: A softball team named the Prairie Dogs is trying to decide where to have its annual team banquet. The players live in three different towns: Buck Springs, Boyce Mills and Ryan Falls. No player wants to travel further than any other. By looking at a map they decide that the banquet should be in one of four places: Morgan City (6,8), Lerchburg (5,9), Flegalville (5,7) or Davistown (8,8). Find by construction which of these towns is equidistant from the three home towns. Then find algebraically the coordinates of the point equidistant from the three home towns. Check your results by finding the straightline distance between your point and the three home towns. The coordinate are: Buck Springs (1,3), Ryan Falls (11,3), and Boyce Mills (7,15).
Thank you for your help!!!!!!
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!



The 3 black points are where they live, and the red points are the 
possible places for the banquet.

It looks as though the red point MC(6,8) Morgan City is the one that's
equidistant from the three black points:



Now we will find algebraically a point which is equidistant from
the three points where they live, WITHOUT ASSUMING that it is
necessarily at Morgan City MC(6,8).

Let the point P(p,q) be a point which is equidistance from all three
black points.  Then using the distance formula

D+=+sqrt%28%28x%5B2%5D-x%5B1%5D%29%5E2%2B%28y%5B2%5D-y%5B1%5D%29%5E2%29

Distance from BS(1,3) to P(p,q)

D+=+sqrt%28%28p-1%29%5E2%2B%28q-3%29%5E2%29

Distance from RF(11,3) to P(p,q)

D+=+sqrt%28%28p-11%29%5E2%2B%28q-3%29%5E2%29

Distance from BM(7,15) to P(p,q)

D+=+sqrt%28%28p-7%29%5E2%2B%28q-15%29%5E2%29

Setting the first two distances equal:

sqrt%28%28p-1%29%5E2%2B%28q-3%29%5E2%29=+sqrt%28%28p-11%29%5E2%2B%28q-3%29%5E2%29

Square both sides:

%28p-1%29%5E2%2B%28q-3%29%5E2=+%28p-11%29%5E2%2B%28q-3%29%5E2

%28p-1%29%5E2%2Bcross%28%28q-3%29%5E2%29=+%28p-11%29%5E2%2Bcross%28%28q-3%29%5E2%29

%28p-1%29%5E2=+%28p-11%29%5E2

p%5E2-2p%2B1=+p%5E2-22p%2B121

cross%28p%5E2%29-2p%2B1=+cross%28p%5E2%29-22p%2B121

-2p%2B1=+-22p%2B121

20p=120

p=6

Setting the first and third distances equal:

sqrt%28%28p-1%29%5E2%2B%28q-3%29%5E2%29+=+sqrt%28%28p-7%29%5E2%2B%28q-15%29%5E2%29

Substituting p=6

sqrt%28%286-1%29%5E2%2B%28q-3%29%5E2%29+=+sqrt%28%286-7%29%5E2%2B%28q-15%29%5E2%29

Squaring both sides:

%286-1%29%5E2%2B%28q-3%29%5E2+=+%286-7%29%5E2%2B%28q-15%29%5E2

5%5E2%2B%28q-3%29%5E2+=+%28-1%29%5E2%2B%28q-15%29%5E2

25%2B%28q-3%29%5E2+=+1%2B%28q-15%29%5E2

25%2Bq%5E2-6q%2B9+=+1%2Bq%5E2-30q%2B225

34%2Bq%5E2-6q+=+226%2Bq%5E2-30q

34%2Bcross%28q%5E2%29-6q+=+226%2Bcross%28q%5E2%29-30q

34-6q+=+226-30q
24q+=+192
q=8

So the point which is equidistant from the three towns
where the players live is P(p,q) or P(6,8), and we see that
this really is the point at Morgan City MC(6,8).

Edwin