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

Question

GEOMETRY.jpg 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.

vandyke1234 · Answer

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? :cry:

)

)

MathRevolution · Answer

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.

bethebest · 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??

chetan2u · Answer

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

MathRevolution · Answer

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?."

Gnpth · Answer

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

chetan2u · Answer

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

niks18 · Answer

I guess this question is similar to the one already discussed here - https://gmatclub.com/forum/if-a-smaller ... fl=similar

Bunuel · Answer

Merged the topics. Thank you.

generis · Answer

gggggg.jpg 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

bumpbot · Answer

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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

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