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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.

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

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