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# In the circle above, three right angles have vertices at the center of

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Math Expert
Joined: 02 Sep 2009
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In the circle above, three right angles have vertices at the center of  [#permalink]

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10 Jan 2019, 02:59
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Difficulty:

25% (medium)

Question Stats:

95% (01:26) correct 5% (01:01) wrong based on 28 sessions

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In the circle above, three right angles have vertices at the center of the circle. If the radius of the circle is 8, what is the combined area of the shaded regions?

A. 8π
B. 9π
C. 12π
D. 13π
E. 16π

Attachment:

2019-01-10_1358.png [ 22.17 KiB | Viewed 415 times ]

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Re: In the circle above, three right angles have vertices at the center of  [#permalink]

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10 Jan 2019, 06:07
In the circle above, three right angles have vertices at the center of the circle. If the radius of the circle is 8, what is the combined area of the shaded regions?

total 360 degrees are included in the circle, and in the figure non-shaded region is $$3*90 = 270 degrees$$

Area of the shaded region = 90 degrees

Area of the sector = $$(x/360)*Pi*r^2$$
= $$(90/360)*Pi*8*8$$ --- radius given as 8
= $$16*PI$$

Option E is correct
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Re: In the circle above, three right angles have vertices at the center of  [#permalink]

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10 Jan 2019, 06:27
r = 8,

Area of shaded region will be 3 * $$\frac{theta}{360}$$* π $$r^2$$ --------------------(1)

At the center of a circle, sum of the angles = $$360^o$$

Sum of remaining angles will be 360 - 270 = 90
So each shaded region will have an angle as$$\frac{90}{3}$$

Substituting this value in (1)
=$$\frac{3}{3}$$* $$\frac{90}{360}$$* π $$8^2$$

=16π

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Re: In the circle above, three right angles have vertices at the center of  [#permalink]

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10 Jan 2019, 06:51

Solution

Given:
• A circle as shown in the figure

To find:
• The combined area of the shaded regions

Approach and Working:
• Angle of each unshaded sector = 90 degrees
o Thus, area of each unshaded sector = $$\frac{90}{360} * ᴨ * r^2 = ᴨ/4 * 8^2 = 16ᴨ$$

• Area of all three unshaded sectors combined = 3 * 16ᴨ = 48ᴨ
• Area of the circle = $$ᴨ * r^2 = 64ᴨ$$

Therefore, area of shaded region = 64ᴨ - 48ᴨ = 16ᴨ

Hence, the correct answer is Option E

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Re: In the circle above, three right angles have vertices at the center of  [#permalink]

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10 Jan 2019, 07:18
Bunuel wrote:

In the circle above, three right angles have vertices at the center of the circle. If the radius of the circle is 8, what is the combined area of the shaded regions?

A. 8π
B. 9π
C. 12π
D. 13π
E. 16π

Attachment:
2019-01-10_1358.png

total angle whose area is to be determined = 360-270 = 90
90/360 * 2 * pi * 8*8
= 16 pi
IMO E
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Re: In the circle above, three right angles have vertices at the center of   [#permalink] 10 Jan 2019, 07:18
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