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In circle O above, if POQ is a right triangle and radius OP = 2, what

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Bunuel
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In circle O above, if POQ is a right triangle and radius OP = 2, what [#permalink] New post   15 Mar 2018, 23:44
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In circle O above, if POQ is a right triangle and radius OP = 2, what is the area of the shaded region?


(A) \(4\pi – 2\)

(B) \(4\pi — 4\)

(C) \(2\pi – 2\)

(D) \(2\pi — 4\)

(E) \(\pi - 2\)


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2018-03-16_1011.png
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Re: In circle O above, if POQ is a right triangle and radius OP = 2, what [#permalink] New post   15 Mar 2018, 23:44
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Bunuel wrote:

In circle O above, if POQ is a right triangle and radius OP = 2, what is the area of the shaded region?


(A) \(4\pi – 2\)

(B) \(4\pi — 4\)

(C) \(2\pi – 2\)

(D) \(2\pi — 4\)

(E) \(\pi - 2\)


Show Spoiler
Attachment:
2018-03-16_1011.png


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In circle O above, if POQ is a right triangle and radius OP = 2, what [#permalink] New post   16 Mar 2018, 01:01
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Bunuel wrote:

In circle O above, if POQ is a right triangle and radius OP = 2, what is the area of the shaded region?


(A) \(4\pi – 2\)

(B) \(4\pi — 4\)

(C) \(2\pi – 2\)

(D) \(2\pi — 4\)

(E) \(\pi - 2\)


Show Spoiler
Attachment:
2018-03-16_1011.png


Area of shaded region = Area of Quarter Circle - Area of triangle POQ

Area of shaded region = (1/4)π2^2 - (1/2)*2*2 = π-2

Answer: option E
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In circle O above, if POQ is a right triangle and radius OP = 2, what [#permalink] New post   16 Mar 2018, 07:56
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Bunuel wrote:

In circle O above, if POQ is a right triangle and radius OP = 2, what is the area of the shaded region?


(A) \(4\pi – 2\)

(B) \(4\pi — 4\)

(C) \(2\pi – 2\)

(D) \(2\pi — 4\)

(E) \(\pi - 2\)

Area of shaded region = (Sector area) - (Triangle Area)

• Sector area as fraction of the circle's area

The key to these kinds of problems often is:
"The sector is what fraction of the circle?"

Use the sector's central angle (given, = 90°) to find that fraction

\(\frac{Part}{Whole}=\frac{SectorAngle}{360°}=\frac{90}{360}=\frac{1}{4}=\frac{SectorArea}{CircleArea}\)

Area of sector = \(\frac{1}{4}\) area of circle

• Area of circle and area of sector

Circle area, r = 2: \(πr^2 = 4π\)
Sector area: \(\frac{4π}{4}\)= \(π\)

• Area of triangle

Area of triangle with radii as sides (s = b and h):
\(\frac{s*s}{2} = \frac{4}{2}\) = \(2\)

• Area of shaded region
= (Area of sector) - (Area of triangle)

Area of shaded region = \(π - 2\)

Answer E
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Re: In circle O above, if POQ is a right triangle and radius OP is 2, wha [#permalink] New post   22 Mar 2022, 05:53
\(\frac{90}{360} = \frac{x}{π2^2}\)

\(\frac{1}{4} = \frac{x}{4π}\)

\(x = π\)

Area of Triangle = \(\frac{1}{2}\) * 2 * 2 =2

Shaded region = Sector Area - Area of traingle
= \(π - 2\)

Answer E
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Re: In circle O above, if POQ is a right triangle and radius OP is 2, wha [#permalink] New post   22 Mar 2022, 09:31
Bunuel wrote:

In circle O above, if ΔPOQ is a right triangle and radius OP is 2, what is the area of the shaded region?

A. 4π− 2
B. 4π− 4
C. 2π− 2
D. 2π− 4
E. π – 2


Show Spoiler
Attachment:
2019-01-10_1324.png

\(\frac{π2^2}{4} - \frac{1*2*2}{2}\)

\(=π – 2\), Answer must be (E)
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In circle O above, if POQ is a right triangle and radius OP is 2, wha [#permalink] New post   22 Mar 2022, 09:31
Bunuel wrote:

In circle O above, if ΔPOQ is a right triangle and radius OP is 2, what is the area of the shaded region?

A. 4π− 2
B. 4π− 4
C. 2π− 2
D. 2π− 4
E. π – 2


Show Spoiler
Attachment:
2019-01-10_1324.png

1/4th Area of Circle - Area of Triangle

\(\frac{π2^2}{4} - \frac{1*2*2}{2}\)

\(=π – 2\), Answer must be (E)
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In circle O above, if POQ is a right triangle and radius OP is 2, wha [#permalink]
22 Mar 2022, 09:31
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