If a smaller circle is inscribed in an equilateral triangle and a lage

I am getting the answer as 4:1

BO :AO = 2:1, Let AO = x, and therefore BO= 2x. They have asked for the ratio of the area of the larger circle to the area of the smaller circle = pi (2x)^2/ pi x^2 = 4:1.

What am I getting wrong??

Hi,
the answer will be 4:1 ..
OR please change the ratio to smaller: larger..
How can larger be less than smaller

Sorry, you are actually right. The question was supposed to be "what is the ratio of the smaller circle’s area to the larger circle’s area?."

The triangle in a diagram is equilateral. The smaller circle is tangent to all three sides of the triangle. The larger circle passes through all three vertices of the triangle. What is the ratio of the area of the smaller circle to the area of the larger circle?

A. 1/8

B. 1/16

C. 1/2

D. 1/4

E. 1/32

Hi..

Draw an altitude of the equilateral triangle..
They will meet at the centre of the incircle and at a point 1/3 of altitude say a, so a/3..
This is the radius of incircle.
When you look at the altitude, the remaining 2/3 *a is nothing but the altitude of outer circle..
So ratio of radius is 1/2..
So ratio of area will become (1/2)^2=1/4

D

I guess this question is similar to the one already discussed here -

https://gmatclub.com/forum/if-a-smaller ... fl=similar

Merged the topics. Thank you.

The equilateral triangle can be used for a quick answer.

An equilateral triangle can be divided, by its three medians, into 6 equal 30-60-90 triangles. Use one of them.

1) Draw two lines

Drop an altitude from B to the base of the triangle, to X.

Then draw a line between O and the triangle's vertex on the right, to Y.

2) Assign a side length to OX, and derive OY from 30-60-90 right triangle properties

Triangle OXY is a 30-60-90 right triangle, with sides in ratio \(x: x\sqrt{3}: 2x\)

OX is the small circle's radius
OY is the large circle's radius

Assign a value to OX*: let OX = 2

By properties of a 30-60-90 right triangle, if OX = 2, OY = 4

3) Find areas of circles, then the ratio needed

Area of small circle: \(\pi*r^2 = 4\pi\)
Area of large circle: \(\pi*r^2 = 16\pi\)

Ratio of small circle's area to large circle's area?

\(\frac{4\pi}{16\pi} = \frac{1}{4} = 1:4\)

ANSWER D

*Or let OX = \(x\). Then OY = \(2x\)

Small circle's area: \(x^2\pi\)
Large circle's area: \(4x^2\pi\)

Ratio of small to large areas is \(1:4\)

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