MathRevolution wrote:

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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

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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 = 43) 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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