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

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

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