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# Coordinates

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The area of equilateral triangle inscribed in the circle x^2+y^2-4x+8y+11=0 is of the form a*sqrt(b)/c.  Find a+b+c.

May 8, 2022

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To find the circumcenter, we first need to know the radius of the circle.

We can rewrite the circle equation to: $$x^2-4x+y^2+8y=-11$$

We can now complete the square, and we get: $$(x-2)^2+(y+4)^2=4+16-11$$

The left hand is equal to 9, meaning the radius is 3.

We now know the circumcenter is 3.

The equation for the circumcenter is: $$\large{{abc} \over {4[ABC]}}$$

But, we have an equilateral triangle, so $$a = b= c$$. This means we can substitute both b and c for a.

This gives us the equation: $$\large{{a^3} \over 4[ABC] } = 3$$

The formula for the area of an equilateral triangle is $${\sqrt 3 \over 4} s^2$$

We can substitute this in, and we get: $${{a^3} \over {{4 \sqrt3 \over 4} a^2}} = 3$$

We can simplify and divide by $$a^2$$ to get: $$\large{a \over \sqrt3} = 3$$

This means that the side of the triangle is $$3 \sqrt 3$$

Can you find the area from here?

Hint: Use the formula for the area of the triangle

May 8, 2022