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.
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
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,
[tex]s=\frac{a+b+c}{2}[/tex]
Where a, b and c are the length of sides.
[tex]s=\frac{48.5+76.4+72.6}{2}[/tex]
[tex]s=98.75[/tex]
The formula to find the radius of the circle pond in the triangle is given below,
[tex]r=\sqrt{\frac{(s-a)(s-b)(s-c)}{s}}[/tex]
[tex]r=\sqrt{\frac{(98.75-76.4)(98.75-72.6)(98.75-48.5)}{98.75} }[/tex]
[tex]r=\sqrt{297.4}[/tex]
[tex]r=17.25[/tex]
The radius of the circle is 17.25 m.
[tex]D=2r[/tex]
[tex]D=2(17.25)[/tex]
[tex]D=34.5[/tex]
Therefore, the diameter of circle is 34.5 m.
