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The hypotenuse of right triangle ABC is the diameter of the circle. If

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Bunuel
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The hypotenuse of right triangle ABC is the diameter of the circle. If [#permalink] New post   17 Nov 2020, 01:15
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The hypotenuse of right triangle ABC is the diameter of the circle. If the diameter of the circle is 18, what is the area of the shaded region?


A. \(81\pi - 81\sqrt{3}\)

B. \(81\pi - \frac{81\sqrt{3}}{2}\)

C. \(81\pi - 27\sqrt{3}\)

D. \(81\pi - \frac{81\sqrt{3}}{4}\)

E. \(81\pi - \frac{81\sqrt{3}}{5}\)


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Re: The hypotenuse of right triangle ABC is the diameter of the circle. If [#permalink] New post   17 Nov 2020, 02:09
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Area of shaded region= Area of circle - Area of right triangle
=> ((pi) x 9 x 9) - (1/2 x base x height)

Traingle's
Base= 18 cos60 = 18 x (1/2) = 9
Height= 18 cos30 = 18 x (\sqrt{3} /2) = 9 \sqrt{3}

=> (81 pi) - (1/2 x 9 x 9 \sqrt{3}
=> \(81\pi - \frac{81\sqrt{3}}{2}\)

ANSWER B



Bunuel wrote:

The hypotenuse of right triangle ABC is the diameter of the circle. If the diameter of the circle is 18, what is the area of the shaded region?


A. \(81\pi - 81\sqrt{3}\)

B. \(81\pi - \frac{81\sqrt{3}}{2}\)

C. \(81\pi - 27\sqrt{3}\)

D. \(81\pi - \frac{81\sqrt{3}}{4}\)

E. \(81\pi - \frac{81\sqrt{3}}{5}\)


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Re: The hypotenuse of right triangle ABC is the diameter of the circle. If [#permalink] New post   17 Nov 2020, 03:20
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IMO: B

Traingle's is of 30,60,90 deg property hence Side will be in the ration of 1:Sqrt3:2
therefore Base= 18 x (1/2) = 9
Height= 18 x (sqrt3)/2 = 9 x (sqrt3)
Therefore are of triangle = 1/2 x base x height = 1/2 x 9 x 9 x (sqrt3)= 1/2 x81 x (sqrt3)

Area of shaded region= Area of circle - Area of right triangle
=> ((pi) x 9 x 9) - (1/2 x base x height)
=> 81π−81√3/2

ANSWER B
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Re: The hypotenuse of right triangle ABC is the diameter of the circle. If [#permalink] New post   17 Nov 2020, 17:53
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Given:
Diameter of the circle = 18
Diameter of the circle = Hypotenuse of the triangle

Need:
Area of shaded region = Area of circle - Area of triangle

Area of Circle:
Diameter = 18, so radius = 9 , Area of circle = \(81\pi\)

Area of Triangle:

Triangle = 30-60-90 , so we know the sides are in the ratio x:x\(\sqrt{3}\): 2x

From the diagram, we know that 2x = 18 , so x = 9
We can now calclulate the other 2 sides , height of triangle = 9 and base is equal to 9\(\sqrt{3}\)

Area of the triangle is equal to \((9 * 9\sqrt{3})/2\) = \((81\sqrt{3})/2\)

Area of shaded region = Area of circle - Area of triangle
Area of shaded region = \(81\pi\) - \((81\sqrt{3})/2\)

Ans: B
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Re: The hypotenuse of right triangle ABC is the diameter of the circle. If [#permalink] New post   17 Nov 2020, 19:39
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Just as the area of an equilateral triangle is \(\frac{\sqrt{3}}{4}* a^2\), where a is the side of the equilateral triangle, we can also remember the areas of a 30 - 60 - 90 and a 45 - 45 - 90 triangle in terms of the hypotenuse.

For a 30 - 60 - 90 triangle, the area = \(\frac{\sqrt{3}}{8}* h^2\)

For a 45 - 45 - 90 triangle, the area = \(\frac{1}{4}* h^2\)


Area of the shaded region = Area of the circle (with radius = 9) - Area of the 30-60-90 triangle (with hypotenuse = 18) = \(\pi r^2 - \frac{\sqrt{3}}{8}* h^2\)


Area = \(9^2 \pi - \frac{\sqrt{3}}{8}* 18^2 = 81 \pi - \frac{81\sqrt{3}}{2}\)



Option B

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Re: The hypotenuse of right triangle ABC is the diameter of the circle. If [#permalink] New post   30 Apr 2022, 23:15
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Re: The hypotenuse of right triangle ABC is the diameter of the circle. If [#permalink]
30 Apr 2022, 23:15
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