So inside.
Geometry
Level
2
A circle is inscribed in an equilateral triangle of side 'a' cm. The area of square inscribed in circle is a *a / k . What is k/2?
The answer is 3.
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As the area of a triangle is (sqrt 3)a^2/4, and also , it is (s*a), where s is the semiperimeter and r is the radius the incircle.
s=3a/2
Hence, (sqrt 3)a^2/4 = (3a/2)*r
On solving, We get r=a^2/ (sqrt 3)*2
Therefore, diameter of the circle = a^2/(sqrt3)
Also, we can notice that diameter of the circle is diagonal of the square.
Area of square is (1/2* d^2)
Here, d =a^2/(sqrt3)
Therefore, area of square= a^2/6 = a^2/k
Hence, k = 6 So, k/2=3