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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: x24x+y2+8y=11

 

We can now complete the square, and we get: (x2)2+(y+4)2=4+1611

 

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: abc4[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: a34[ABC]=3

 

The formula for the area of an equilateral triangle is 34s2

 

We can substitute this in, and we get: a3434a2=3

 

We can simplify and divide by a2 to get: a3=3

 

This means that the side of the triangle is 33

 

Can you find the area from here?

 

Hint: Use the formula for the area of the triangle

 May 8, 2022

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