CAT question from geometry equilateral triangle.

1. Geometry

2. Geometry
Two circles are placed in an equilateral triangle as shown in the
figure. What is the ratio of the area of the smaller circle to that of
the equilateral triangle?
(a) π : 36 3
(b) π : 18 3
(c) π : 27 3
(d) π : 42 3

3. Geometry
In-radius of equilateral triangle of side a =
a
2 3
Diameter of larger circle =
a
2 3
Let us say common tangent PQ touches the
two circle at R, center of smaller circle is I.
Two circles are placed in an equilateral triangle as shown in the
figure. What is the ratio of the area of the smaller circle to that of
the equilateral triangle?

4. Geometry
Now, PQ is parallel to BC. AR is perpendicular to PQ.
Triangle PQR is also an equilateral triangle and AORID
is a straight line. (Try to establish each of these
observations. Just to maintain the rigour.)
AD =
3
2
a
RD =
a
3
Two circles are placed in an equilateral triangle as shown in the
figure. What is the ratio of the area of the smaller circle to that of
the equilateral triangle?

5. Geometry
AR =
3
2
a –
a
3
=
3a − 2a
2 3
=
a
2 3
; AR =
1
3
AD.
Radius of smaller circle =
1
3
radius of larger circle
Radius of smaller circle =
1
3
∗
a
2 3
=
a
6 3
Two circles are placed in an equilateral triangle as shown in the
figure. What is the ratio of the area of the smaller circle to that of
the equilateral triangle?

6. Geometry
Area of smaller circle = r2
 
a
6 3
2
=
πa2
108
Area of ∆ =
3
4
a2
Ratio =
πa2
108
:
3
4
a2
 : 27 3. Answer choice (c)
Two circles are placed in an equilateral triangle as shown in the
figure. What is the ratio of the area of the smaller circle to that of
the equilateral triangle?

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