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In the figure above, if radius OA is 8 and the area of right triangle

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In the figure above, if radius OA is 8 and the area of right triangle  [#permalink]

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New post 10 Jan 2019, 01:29
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In the figure above, if radius OA is 8 and the area of right triangle OAB is 32, what is the area of the shaded region?

A. 64π + 32
B. 60π + 32
C. 56π + 32
D. 32π + 32
E. 16π + 32


Attachment:
2019-01-10_1328.png
2019-01-10_1328.png [ 17.67 KiB | Viewed 191 times ]

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Re: In the figure above, if radius OA is 8 and the area of right triangle  [#permalink]

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New post 10 Jan 2019, 02:15
In the figure above, if radius OA is 8 and the area of right triangle OAB is 32, what is the area of the shaded region?

Area of the circle = Pi*8^2
= \(64*Pi\)

Area of the right triangle = 32

Since Angle OAB = 90 degrees
and \(1/2*AB*OA\) = 32 (Given OA = radius of circle = 8)
implies AB = 8
implies AB = OA = 8, OAB is right isosceles triangle with AOB = OBA = 45 degrees

Area of the part of the circle excluding the right triangle = \(((360 - 45)/360)*64*PI\)
= \(56*PI\)
Total shaded region = \(56*Pi + 32\)
Option C is correct
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Re: In the figure above, if radius OA is 8 and the area of right triangle  [#permalink]

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New post 10 Jan 2019, 04:56
Bunuel wrote:
Image
In the figure above, if radius OA is 8 and the area of right triangle OAB is 32, what is the area of the shaded region?

A. 64π + 32
B. 60π + 32
C. 56π + 32
D. 32π + 32
E. 16π + 32


Attachment:
2019-01-10_1328.png



given area of circle = 64pi
and side ab = 0.5*oa*ab= 32
ab= 8
so we can say that oa=ab= 8 and triangel oab is 45:45;90 triangle and angle AOB is 45
so area of side AOC ' c is point on circumference on length ob'
45/360* pi * 8 * 8 = 8 pi

so total area of the figure : 64pi-8pi + 32
; 56pi+32 IMO C
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Re: In the figure above, if radius OA is 8 and the area of right triangle  [#permalink]

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New post 10 Jan 2019, 05:43
Bunuel wrote:
Image
In the figure above, if radius OA is 8 and the area of right triangle OAB is 32, what is the area of the shaded region?

Attachment:
2019-01-10_1328.png


Area of Triangle OAB
=> 1/2 OA * AB = 32
=> AB = 8

From this we can say that triangle OAB is 45-45-90, implying < AOB = 45

To calculate the area of the inner sector(one which doesn't include the triangle OAB), for the formula \(\frac{theta}{360}\)* π \(r^2\)
We can caluclaet the theta by 360-45 = 315
Now, \(\frac{315}{360}\)* π \(8^2\)
=> 56 π

Total Area will be 56 π + 32

Answer C
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Re: In the figure above, if radius OA is 8 and the area of right triangle &nbs [#permalink] 10 Jan 2019, 05:43
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