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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
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If a smaller circle is inscribed in an equilateral triangle and a lage  [#permalink]

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Difficulty:   45% (medium)

Question Stats: 61% (01:45) correct 39% (01:49) wrong based on 80 sessions

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Attachment: GEOMETRY.jpg [ 7.41 KiB | Viewed 12722 times ]

If a smaller circle is inscribed in an equilateral triangle and a lager circle circumscribed about the triangle shown as above figure, what is the ratio of the smaller circle’s area to the larger circle’s area?

A. 1:2
B. 1:√3
C. 1:3
D. 1:4
E. 1:5

* A solution will be posted in two days.

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Re: If a smaller circle is inscribed in an equilateral triangle and a lage  [#permalink]

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With an equilateral triangle, the radius of the incircle is exactly half the radius of the circumcircle.

so, let the radius of the circumcircle be x, then the radius of incircle = x/2

Area of circumcircle/ area of incircle = pi*x^2 / pi* (x/2) ^2
=4/1 (which is not in the answer... am i missing something? )
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Re: If a smaller circle is inscribed in an equilateral triangle and a lage  [#permalink]

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If a smaller circle is inscribed in an equilateral triangle and a lager circle circumscribed about the triangle shown as above figure, what is the ratio of the lager circle’s area to the smaller circle’s area?

A. 1:2
B. 1:√3
C. 1:3
D. 1:4
E. 1:5

-> In the above picture, angle A=90 degrees and ABO=30 degrees, which makes AO:BO=1:2. Since ratio of area=the square of ratio of length, it is (1:2)^2=(1/2)^2=1/4=1:4. Thus, D is the answer.
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Re: If a smaller circle is inscribed in an equilateral triangle and a lage  [#permalink]

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MathRevolution wrote:
If a smaller circle is inscribed in an equilateral triangle and a lager circle circumscribed about the triangle shown as above figure, what is the ratio of the lager circle’s area to the smaller circle’s area?

A. 1:2
B. 1:√3
C. 1:3
D. 1:4
E. 1:5

-> In the above picture, angle A=90 degrees and ABO=30 degrees, which makes AO:BO=1:2. Since ratio of area=the square of ratio of length, it is (1:2)^2=(1/2)^2=1/4=1:4. Thus, D is the answer.

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??
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Joined: 02 Aug 2009
Posts: 7954
Re: If a smaller circle is inscribed in an equilateral triangle and a lage  [#permalink]

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MathRevolution wrote:
If a smaller circle is inscribed in an equilateral triangle and a lager circle circumscribed about the triangle shown as above figure, what is the ratio of the lager circle’s area to the smaller circle’s area?

A. 1:2
B. 1:√3
C. 1:3
D. 1:4
E. 1:5

-> In the above picture, angle A=90 degrees and ABO=30 degrees, which makes AO:BO=1:2. Since ratio of area=the square of ratio of length, it is (1:2)^2=(1/2)^2=1/4=1:4. Thus, D is the answer.

Hi,
the answer will be 4:1 ..
OR please change the ratio to smaller: larger..
How can larger be less than smaller
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Re: If a smaller circle is inscribed in an equilateral triangle and a lage  [#permalink]

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MeghaP wrote:
MathRevolution wrote:
If a smaller circle is inscribed in an equilateral triangle and a lager circle circumscribed about the triangle shown as above figure, what is the ratio of the lager circle’s area to the smaller circle’s area?

A. 1:2
B. 1:√3
C. 1:3
D. 1:4
E. 1:5

-> In the above picture, angle A=90 degrees and ABO=30 degrees, which makes AO:BO=1:2. Since ratio of area=the square of ratio of length, it is (1:2)^2=(1/2)^2=1/4=1:4. Thus, D is the answer.

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??

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?."
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The triangle in a diagram is equilateral. The smaller circle is  [#permalink]

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File comment: Problem Capture.PNG [ 9.03 KiB | Viewed 11420 times ]

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
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Re: The triangle in a diagram is equilateral. The smaller circle is  [#permalink]

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Gnpth wrote:
Attachment:
Capture.PNG

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
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Re: The triangle in a diagram is equilateral. The smaller circle is  [#permalink]

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1
Gnpth wrote:
Attachment:
Capture.PNG

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

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

https://gmatclub.com/forum/if-a-smaller ... fl=similar
Math Expert V
Joined: 02 Sep 2009
Posts: 58320
Re: If a smaller circle is inscribed in an equilateral triangle and a lage  [#permalink]

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niks18 wrote:
Gnpth wrote:
Attachment:
Capture.PNG

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

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.
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If a smaller circle is inscribed in an equilateral triangle and a lage  [#permalink]

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MathRevolution wrote:
Attachment:
The attachment GEOMETRY.jpg is no longer available

If a smaller circle is inscribed in an equilateral triangle and a lager circle circumscribed about the triangle shown as above figure, what is the ratio of the smaller circle’s area to the larger circle’s area?

A. 1:2
B. 1:√3
C. 1:3
D. 1:4
E. 1:5

Attachment: gggggg.jpg [ 22.28 KiB | Viewed 11221 times ]

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?

$$\frac{4\pi}{16\pi} = \frac{1}{4} = 1:4$$

*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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Re: If a smaller circle is inscribed in an equilateral triangle and a lage  [#permalink]

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_________________ Re: If a smaller circle is inscribed in an equilateral triangle and a lage   [#permalink] 06 Feb 2019, 03:42
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