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    Diameter of the largest circular pond in a triangular garden

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    Find the diameter of the largest circular pond that could fit in a triangular garden with vertices at (18,54), (-27,36), and (27,-18), where a unit reprsents 1m.

    © BrainMass Inc. brainmass.com December 24, 2021, 5:14 pm ad1c9bdddf
    https://brainmass.com/math/geometry-and-topology/diameter-of-the-largest-circular-pond-in-a-triangular-garden-37161

    SOLUTION This solution is FREE courtesy of BrainMass!

    The in center is the center of the incircle, the inscribed circle of the triangle

    The vertices of the triangle are ABC and the sides of the triangle are a,b,c

    Point x cordinate y cordinate
    A 18 54
    B -27 36
    C 27 -18

    a = | BC|= 76.3675 ={(-27)-(27)}^2+{(36)-(-18)}^2
    b = | AC|= 72.5603 ={(27)-(18)}^2+{(-18)-(54)}^2
    c = | AB|= 48.4665 ={(18)-(-27)}^2+{(54)-(36)}^2

    X coordinate of in center = (ax1+bx2+cx3/ ((a+b+c) = 3.6682
    Y coordinate of in center = (ay1+by2+cy3)/ (a+b+c) 29.7051

    The radius of the circle (pond) is the perpendicular distance between the in center and one of the sides of the triangle

    The perpendicular distance d of a point P (x 1, y 1) from the line lx +my +n = 0 is given by
    d =| lx1 +my1 +n|/square root of [(l^2 +m^2)]

    Point x coordinate y coordinate
    A 18 54
    B -27 36
    Equation of AB is
    0.4 x-y+ 46.8 = 0

    l= 0.4
    m= -1
    n= 46.8
    Co-ordinates of in center=
    x1= 3.6682
    y1= 29.7051
    Therefore perpendicular distnce= 17.2346

    This is the radius of the pond
    Diameter = 2 x radius = 34.4692

    Answer: Diameter of the pond= 34.4692 m

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 5:14 pm ad1c9bdddf>
    https://brainmass.com/math/geometry-and-topology/diameter-of-the-largest-circular-pond-in-a-triangular-garden-37161

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