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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
    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.

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