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 = 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
\(1:4\)
_________________
SC Butler has resumed! Get
two SC questions to practice, whose links you can find by date,
here.Choose life.