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

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Math Expert
Joined: 02 Sep 2009
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In circle O above, if POQ is a right triangle and radius OP = 2, what  [#permalink]

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15 Mar 2018, 23:44
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25% (medium)

Question Stats:

85% (01:12) correct 15% (02:01) wrong based on 43 sessions

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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) $$\pi - 2$$

Attachment:

2018-03-16_1011.png [ 9.71 KiB | Viewed 807 times ]

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Re: In circle O above, if POQ is a right triangle and radius OP = 2, what  [#permalink]

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15 Mar 2018, 23:44
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) $$\pi - 2$$

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]

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16 Mar 2018, 01:01
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) $$\pi - 2$$

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

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

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16 Mar 2018, 07:56
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 sector) - (Area of triangle)

Area of shaded region = $$π - 2$$

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In circle O above, if POQ is a right triangle and radius OP = 2, what   [#permalink] 16 Mar 2018, 07:56
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