A circle is made inside an equilateral triangle and inside the circle
A square is formed. The area of the triangle with the area of the square
The ratio will be:
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Step-by-step explanation:
Clearly, centre of triangle , circle and square coincides. (Centroid, incenter of equilateral triangle , orthocenter coincides)
Let a be the length of side of an equilateral triangle.
Area of equilateral triangle A
1
=
4
3
a
2
We have radius of inscribed circle r=
2
3
a×
3
1
=
2
3
a
So, diameter=
3
a
We know that the diameter of the inscribed circle is equal to the diagonal of the square.
So, diagonal of square =
3
a
Let x be the length of side of square.
Length of diagonal of square =x
2
⇒x
2
=
3
a
⇒x=
6
a
Area of square A
2
=x
2
=
6
a
2
Now,
A
2
A
1
=
6
a
2
4
3
a
2
⇒
A
2
A
1
=
2
3
3
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