# Geometry

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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?

• Geometry -

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 (17-x)
AC: y = 3/14 (4x+3)
CA: y = 1/2 (127-20x)

These all intersect at (220/38,213/38)

The radius of the circle is 5√1717/38

(x-220/38)^2 + (y-213/38)^2 = 42925/1444 = 29.726

Ouch! Hard to believe, but the math checks out.