PathFinder007 wrote:
A circle is inscribed in an equilateral triangle, such that the two figures touch at exactly 3 points, one on each side of the triangle. Which of the following is closest to the percent of the area of the triangle that lies within the circle?
A 20
B 45
C 60
D 55
E 77
Assume this as an equilateral triangle with side as a (Yes I know, this isn't an equilateral triangle hence i asked you to assume, this is the only diagram I got from internet and I am too lazy to draw one myself)
Attachment:
File comment: Diagram
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so now area of an equilateral triangle = \(\frac{\sqrt3}{4} * a^2\) -----1
and area of the triangle is also equal to Area of triangle AOC+Area of AOB + Area of BOC = \(\frac{1}{2} * a * r\)( r is height of individual triangle) ---------2
from equation 1 & 2 above
\(\frac{\sqrt3}{4} * a^2 = 3 * \frac{1}{2} * a * r\)
from here we can get the value of a i.e. \(a = 2\sqrt{3} * r\) ---------3
Now, In the question we need to find out \(\frac{Area of circle inscribed}{Area of the equilateral triangle}\)
which is equal to \(\frac{{\pi * r^2}}{{3* 1/2 * (2 \sqrt3 r)^2}}\) ------substituting the value of a from equation 3
=\(\frac{\pi}{{3 * \sqrt3}}\)
\(\approx \frac{3.14}{{3 * 1.72}}\)
\(\approx \frac{3.14}{{3 * 1.72}}\)
\(\approx \frac{2}{3} \approx 0.66\)
Notice that we reduced the numerator by \(0.26\) so our answer is going to be a bit inflated.
Looking at the answer choices, C is the closest. (Notice that there's nothing between 60 and 70 in the options so we can be a little imprecise in this case)
Hence the solution is C