Answer: The diameter of the largest circular pond that could fit in a triangular garden with vertices at (18,54), (-27,36), & (27,-18) is 34.5m.
Explanation:
Let the vertices of the triangle are A(18,54), B(-27,36), & C(27,-18).
Use distance formula to find the length of sides.
![d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28x_2-x_1%29%5E2%2B%28y_2-y_1%29%5E2%7D)
Using the above formula the distance between AB is 48.5, BC is 76.4 and AC is 72.6.
The length of sides are 48.5, 76.4 and 72.6.
Formula to find semi-perimeter is given below,
![s=\frac{a+b+c}{2}](https://tex.z-dn.net/?f=s%3D%5Cfrac%7Ba%2Bb%2Bc%7D%7B2%7D)
Where a, b and c are the length of sides.
![s=\frac{48.5+76.4+72.6}{2}](https://tex.z-dn.net/?f=s%3D%5Cfrac%7B48.5%2B76.4%2B72.6%7D%7B2%7D)
![s=98.75](https://tex.z-dn.net/?f=s%3D98.75)
The formula to find the radius of the circle pond in the triangle is given below,
![r=\sqrt{\frac{(s-a)(s-b)(s-c)}{s}}](https://tex.z-dn.net/?f=r%3D%5Csqrt%7B%5Cfrac%7B%28s-a%29%28s-b%29%28s-c%29%7D%7Bs%7D%7D)
![r=\sqrt{\frac{(98.75-76.4)(98.75-72.6)(98.75-48.5)}{98.75} }](https://tex.z-dn.net/?f=r%3D%5Csqrt%7B%5Cfrac%7B%2898.75-76.4%29%2898.75-72.6%29%2898.75-48.5%29%7D%7B98.75%7D%20%7D)
![r=\sqrt{297.4}](https://tex.z-dn.net/?f=r%3D%5Csqrt%7B297.4%7D)
![r=17.25](https://tex.z-dn.net/?f=r%3D17.25)
The radius of the circle is 17.25 m.
![D=2r](https://tex.z-dn.net/?f=D%3D2r)
![D=2(17.25)](https://tex.z-dn.net/?f=D%3D2%2817.25%29)
![D=34.5](https://tex.z-dn.net/?f=D%3D34.5)
Therefore, the diameter of circle is 34.5 m.