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

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

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02 Mar 2016, 22:58
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45% (medium)

Question Stats:

64% (01:13) correct 36% (01:29) wrong based on 90 sessions

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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 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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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Manager Status: Keep it simple stupid!! Joined: 02 Mar 2016 Posts: 73 Location: India Concentration: General Management, Operations GMAT 1: 680 Q47 V36 GPA: 3.37 Re: If a smaller circle is inscribed in an equilateral triangle and a lage [#permalink] ### Show Tags 03 Mar 2016, 00:24 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? ) _________________ "Give without remembering, take without forgetting" - Kudos Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6028 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: If a smaller circle is inscribed in an equilateral triangle and a lage [#permalink] ### Show Tags 06 Mar 2016, 03:39 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. _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Re: If a smaller circle is inscribed in an equilateral triangle and a lage  [#permalink]

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26 Mar 2016, 23:02
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??
Math Expert
Joined: 02 Aug 2009
Posts: 6548
Re: If a smaller circle is inscribed in an equilateral triangle and a lage  [#permalink]

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26 Mar 2016, 23:42
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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27 Mar 2016, 23:22
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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01 Oct 2017, 11:34
Top Contributor
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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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01 Oct 2017, 12:03
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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01 Oct 2017, 12:09
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
Joined: 02 Sep 2009
Posts: 47978
Re: If a smaller circle is inscribed in an equilateral triangle and a lage  [#permalink]

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01 Oct 2017, 12:15
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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03 Oct 2017, 13:32
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 6832 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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