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A smaller circle is inscribed in a larger circle shown as above figure

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A smaller circle is inscribed in a larger circle shown as above figure [#permalink]

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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. \(\frac{5}{6}\)
[Reveal] Spoiler: OA

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Re: A smaller circle is inscribed in a larger circle shown as above figure [#permalink]

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New post 11 Jul 2017, 19:20
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MathRevolution wrote:
Attachment:
7.11.png


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
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Re: A smaller circle is inscribed in a larger circle shown as above figure [#permalink]

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New post 11 Jul 2017, 23:05
chetan2u wrote:
MathRevolution wrote:
Attachment:
7.11.png


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??
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A smaller circle is inscribed in a larger circle shown as above figure [#permalink]

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New post 12 Jul 2017, 08:07
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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
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.
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Re: A smaller circle is inscribed in a larger circle shown as above figure [#permalink]

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New post 12 Jul 2017, 08:25
chetan2u wrote:
smaller circle passing through the centre and touching the circumference at just one point.
C

Good catch, I was also confused after reading and only by assuming that was able to resolve the case
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Re: A smaller circle is inscribed in a larger circle shown as above figure [#permalink]

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New post 13 Jul 2017, 01:06
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==> Square of the ratio of the length=ratio of the area. The diameter of the smaller circle is equal to the radius of the larger circle, so the ratio of the length becomes 1:2. Then, the ratio of the area becomes 1^2:2^2=1:4. Thus, if the area of the smaller circle=k, the area of the larger circle=4k. Therefore, the area of the region shaded:the area of the larger circle=(4k-k):4k=3:4=¾.

The answer is C.
Answer: C
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Re: A smaller circle is inscribed in a larger circle shown as above figure [#permalink]

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New post 03 Sep 2017, 00:25
Please correct the figure, I got it wrong just because of faulty figure.
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Re: A smaller circle is inscribed in a larger circle shown as above figure [#permalink]

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New post 03 Sep 2017, 10:04
This question is a complete waste of time with current attachment

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Re: A smaller circle is inscribed in a larger circle shown as above figure   [#permalink] 03 Sep 2017, 10:04
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