Geometry
posted by Rachel .
A gardener wants the three rosebushes in her garden to be watered by a rotating water sprinkler. The gardener draws a diagram of the garden using a grid in which each unit represents 1 ft. The rosebushes 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 gardener place the sprinkler? What equation describes the boundary of the circular region that the sprinkler will cover?

the center of the circle will be at the intersection of the perpendicular bisectors of two of the line segments joining the points. If we label the points
A:(1,3)
B:(5,11)
C:(11,4)
then we have
AB: midpoint=(3,7) slope=2
BC: midpoint=(8,15/2) slope=7/6
CA: midpoint=(6,7/2) slope=1/10
so, the equations of the perpendicular bisectors are
AB: y = 1/2 (17x)
AC: y = 3/14 (4x+3)
CA: y = 1/2 (12720x)
These all intersect at (220/38,213/38)
The radius of the circle is 5√1717/38
(x220/38)^2 + (y213/38)^2 = 42925/1444 = 29.726
Ouch! Hard to believe, but the math checks out.
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