Adityagmatclub wrote:

chetan2u wrote:

MathRevolution wrote:

A smaller circle is inscribed in a larger circle shown as above figure. If the smaller circle passes through the center of the larger circle, what is the ratio of the area of the region shaded to the area of the larger circle?

A. \(\frac{1}{2}\)

B. \(\frac{2}{3}\)

C. \(\frac{3}{4}\)

D. \(\frac{4}{5}\)

E. \(\frac{5}{6}\)

Hi,

The figure is incorrect and does not represent a sketch that is intended by Q.

Also there can be various smaller circles that can satisfy the Q so the Q must be meaning :-smaller circle passing through the centre and touching the circumference at just one point.Solution:-

This circle will have the DIAMETER equal to the RADIUS of larger circle...

Area of larger circle =πr^2...

Area of smaller circle=\(π(\frac{r}{2)}^2=π\frac{r^2}{4}\)..

Thus area of shaded region =\(πr^2-π\frac{r^2}{4}=\frac{3πr^2}{4}\)...

Thus ratio =3/4

C

How diameter of small circle equals to radius of larger circle??

chetan2u , thanks for clarifying. I thought the same, but waited for an expert . . .

Adityagmatclub , see my figure and solution immediately below where smaller circle passes through center of larger circle and smaller circle touches larger circle's circumference at just one point.

Attachment:

circle--in-circle.png [ 2.34 KiB | Viewed 897 times ]
Similar to

chetan2u 's analysis: Disregard the figure. Rely instead on the phrase "is inscribed in," see my figure above, where the smaller circle is completely inside the larger circle, which is what I understand "inscribed" to mean.

Smaller circle passes through larger circle's center and touches larger circle's circumference at just one point.

Shaded area = Area of large circle - area of smaller circle.

Let radius of larger circle be 4.

Radius of larger circle is diameter of small circle, and d = 2r, so radius of small circle is: d = 4 = 2r means r = 2.

Area of large circle = \(\pi*r^2\) = \(\pi*4^2\) = 16\(\pi\)

Area of small circle (r is 2) = 4\(\pi\)

Large - small = 16\(\pi\) - 4\(\pi\) = 12\(\pi\) --> is the area of the shaded portion

Ratio of area of shaded portion to area of larger circle is \(\frac{12\pi}{16\pi}\) = \(\frac{3}{4}\)

Answer C

Hope it helps.

_________________

At the still point, there the dance is. -- T.S. Eliot

Formerly genxer123