A gardner wants the rosebushes in her garden to be water by a rotating water sprinkler. The gardner draws a diagram of the garden using a grid in which each unit represents 1 ft. The rosebushed are at (1,3), (5,11), and (11,4). She wants to position the sprinkler at a point equidistant from each rosebush. Where should the gardner place the sprinkler? What equation describes the boundary of the circular region that he sprinkler will cover?

Question

anonymous · Answer

I believe you use a midpoint formula being m=(x1+x2/2, y1+y2/2) but that didnt really work so i need help with what formula to use
i already drew it on graph paper.

anonymous · Answer

i know it's coordinate geometry

anonymous · Answer

I think you might need to set up a system of equations using the distance formula. Let d be the distance between the sprinkler and the rosebushes, and (x, y) be the coordinates of the sprinkler. Then the three equations for the system are

$$d=\sqrt{(x-1)^2+(y-3)^2}$$

$$d=\sqrt{(x-5)^2+(y-11)^2}$$

$$d=\sqrt{(x-11)^2+(y-4)^2}$$

anonymous · Answer

Set any two of the equations equal to each other to eliminate d.

anonymous · Answer

Or you could draw a triangle between the three points and find the intersection perpendicular bisectors

anonymous · Answer

it's impossible so i gave up and gave it to my teacher but thank you.