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

Question

https://gmatclub.com/forum/download/file.php?id=58610 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} Project PS Butler Subscribe to get Daily Email - Click Here | Subscribe via RSS - RSS 2020-10-27_14-09-11.png

Pankaj0901 · Answer

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

mdsaddamforgmat · Answer

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

akadiyan · Answer

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

CrackverbalGMAT · Answer

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

Arun Kumar

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

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

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