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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.

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