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In the figure above, the measure of angle AOB is 120 degrees. If the

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Bunuel
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In the figure above, the measure of angle AOB is 120 degrees. If the [#permalink] New post   17 Jan 2019, 04:20
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In the figure above, the measure of angle AOB is 120 degrees. If the radius of OB is x, then arc AB is what fraction of the area of sector AOB, in terms of x?


A π/x

B √3/x

C 2/x

D 3/x

E 4/x


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2019-01-17_1517.png
2019-01-17_1517.png [ 9.6 KiB | Viewed 6045 times ]
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C
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Re: In the figure above, the measure of angle AOB is 120 degrees. If the [#permalink] New post   17 Jan 2019, 04:31
Bunuel wrote:

In the figure above, the measure of angle AOB is 120 degrees. If the radius of OB is x, then arc AB is what fraction of the area of sector AOB, in terms of x?


A π/x

B √3/x

C 2/x

D 3/x

E 4/x


Show Spoiler
Attachment:
2019-01-17_1517.png


Length of arc \(AB=2πx*(\frac{120}{360})=2πx*k\) (where \(k=\frac{120}{360}\))

Area of sector \(AOB=πx^2*(\frac{120}{360})\)=\(πx^2*k\)

Now, measure of arc AB: Measure of area of sector AOB=\(2πx*k:πx^2*k\)=\(2:x\)

Ans. (C)
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Re: In the figure above, the measure of angle AOB is 120 degrees. If the [#permalink] New post   17 Jan 2019, 04:32
Imo C

2/x

2πX 120/360 / πx2 120/360

2/x

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Re: In the figure above, the measure of angle AOB is 120 degrees. If the [#permalink] New post   17 Jan 2019, 05:34
Bunuel wrote:

In the figure above, the measure of angle AOB is 120 degrees. If the radius of OB is x, then arc AB is what fraction of the area of sector AOB, in terms of x?


A π/x

B √3/x

C 2/x

D 3/x

E 4/x


Show Spoiler
Attachment:
2019-01-17_1517.png


Just a quick question, for this figure why are we not considering the outer sector with theta as \(240^o\)

making the answer as E.
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In the figure above, the measure of angle AOB is 120 degrees. If the [#permalink] New post   17 Jan 2019, 07:11
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Solution


Given:
    • Angle AOB = 120 degrees
    • Radius, OB = x

To find:
    • Length of arc, AB/area of sector AOB

Approach and Working:
    • Length of arc, AB = \(\frac{120}{360} * 2ᴨ * x = \frac{2ᴨx}{3}\)
    • Area of sector AOB = \(\frac{120}{360} * ᴨ * x^2 = \frac{ᴨx^2}{3}\)

Therefore, the answer is \(\frac{2ᴨx}{3}/\frac{ᴨx^2}{3} = \frac{2}{x}\)

Hence, the correct answer is Option C

Answer: C

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Re: In the figure above, the measure of angle AOB is 120 degrees. If the [#permalink] New post   17 Jan 2019, 08:35
Bunuel wrote:

In the figure above, the measure of angle AOB is 120 degrees. If the radius of OB is x, then arc AB is what fraction of the area of sector AOB, in terms of x?


A π/x

B √3/x

C 2/x

D 3/x

E 4/x



Show Spoiler
Attachment:
2019-01-17_1517.png



area of sector : 120/360 * pi * x2 and arc AB is 120/360 * 2 * pi * x

ratio would give us
2/x IMO C
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Re: In the figure above, the measure of angle AOB is 120 degrees. If the [#permalink] New post   21 Jan 2019, 19:11
Expert Reply
Bunuel wrote:

In the figure above, the measure of angle AOB is 120 degrees. If the radius of OB is x, then arc AB is what fraction of the area of sector AOB, in terms of x?


A π/x

B √3/x

C 2/x

D 3/x

E 4/x


Show Spoiler
Attachment:
2019-01-17_1517.png


Since the radius is x, the area is πx^2, and since AB corresponds to a 120 degree angle, the area of sector AOB is 120/360 * πx^2 = 1/3 * πx^2 = (πx^2)/3.

Next we determine the circumference of the entire circle and the arclength of arc AB.

Circumference = 2xπ, so arc AB is 120/360 * 2xπ = 1/3 * 2xπ = 2xπ/3.

Finally arc AB is (2xπ/3)/(πx^2/3) = 6xπ/3πx^2 = 2/x of the area of sector AOB.

Answer: C
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Re: In the figure above, the measure of angle AOB is 120 degrees. If the [#permalink] New post   24 Jan 2019, 13:34
Bunuel wrote:

In the figure above, the measure of angle AOB is 120 degrees. If the radius of OB is x, then arc AB is what fraction of the area of sector AOB, in terms of x?


A π/x

B √3/x

C 2/x

D 3/x

E 4/x


If x is the radius that means \(2pi * x\) is the perimeter and \(x^2 pi\) is the area

ARC AB is \(\frac{1}{3} of the total perimeter\) and area of AOB is also \(\frac{1}{3}\) of the total area.

This means that perimeter : area = (2pi*x/3) : (x^2*pi/3) = \(\frac{2}{x}\)
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Re: In the figure above, the measure of angle AOB is 120 degrees. If the [#permalink]
24 Jan 2019, 13:34
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