A circle has an area equal to four times its perimeter. The maximum possible area of a triangle inscribed in the circle can be written as a√b where a and b are integers and b is not divisible by the square of any prime. What is a + b?
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Answer:
Area of circle = πr²
Perimeter (circumference) = 2πr
∴ πr² = 4(2πr)
→ πr² = 8πr
→ r = 8 units.
Here, r is the radius. Radius is 8 units.
Here, there is a very interesting fact. The largest possible triangle that can be inscribed on a circle is an equilateral triangle.
It has an area of a√b.
Now, there is a very interesting formula:
Area of equilateral triangle inscribed in a circle:
3/4 × r² × √3
r = 8.
→ 3/4 × 8² × √3
→ 48√3.
Another formula, A = √3 /4 × a²
Therefore,
48√3 = √3 /4 × a²
48 × 4 = a²
a = √192
a = 8√3
Then, b = (1/2) * √3 * a
b = (1/2) * √3 * 8√3
b = 12.
a+b = 8√3 + 12.
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