SOLUTION: Find the diameter of the largest circular pond that could fit in a triangular garden with verticies at (18,54),(-27,36),(27,-18, where a unit represents 1m

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Question 1126088: Find the diameter of the largest circular pond that could fit in a triangular garden with verticies at (18,54),(-27,36),(27,-18, where a unit represents 1m
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!



first use distance formula to find the length of the sides:
a triangular garden with vertices at
A=(18,54),
B=(-27,36),
C=(27,-18)

find AB:
Solved by pluggable solver: Distance Between 2 points
The distance formula is sqrt%28%28%28x%5B2%5D-x%5B1%5D%29%5E2%29%2B%28%28y%5B2%5D-y%5B1%5D%29%5E2%29%29. Plug in the numbers,
sqrt%28%28%28-27-%2818%29%29%5E2%29%2B%28%2836-%2854%29%29%5E2%29%29
sqrt%28-45%5E2%2B-18%5E2%29 The distance is 48.4664832642105.




AB=c ->c=48.5

find BC:
Solved by pluggable solver: Distance Between 2 points
The distance formula is sqrt%28%28%28x%5B2%5D-x%5B1%5D%29%5E2%29%2B%28%28y%5B2%5D-y%5B1%5D%29%5E2%29%29. Plug in the numbers,
sqrt%28%28%28-27-%2827%29%29%5E2%29%2B%28%2836-%28-18%29%29%5E2%29%29
sqrt%28-54%5E2%2B54%5E2%29 The distance is 76.3675323681471.



BC=a ->a=76.4


find AC:
Solved by pluggable solver: Distance Between 2 points
The distance formula is sqrt%28%28%28x%5B2%5D-x%5B1%5D%29%5E2%29%2B%28%28y%5B2%5D-y%5B1%5D%29%5E2%29%29. Plug in the numbers,
sqrt%28%28%2818-%2827%29%29%5E2%29%2B%28%2854-%28-18%29%29%5E2%29%29
sqrt%28-9%5E2%2B72%5E2%29 The distance is 72.5603197346869.



AC=b ->b=72.6

Here is a formula in terms of the three sides:
If the sides have length a,b, c, we define the semi-perimeter s to be half their sum, so s+=+%28a%2Bb%2Bc%29%2F2.
s+=+%2876.4%2B72.6%2B48.5%29%2F2
s+=+98.75

Given this, the radius is given using the following:
+r%5E2+=+%28s+-+a%29%2A%28s+-+b%29%2A%28s+-+c%29+%2F+s
+r%5E2+=+%2898.75+-+76.4%29%2A%2898.75+-+72.6%29%2A%2898.75+-+48.5%29+%2F+98.75
+r%5E2+=+297.4
Take the square root of this expression to find r.
+r+=+sqrt%28297.4%29
+r+=+17.25 meters

then, the diameter is d=34.5 meters