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If in the figure above, semicircles X and Y have radii of 6 and 8 feet

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If in the figure above, semicircles X and Y have radii of 6 and 8 feet  [#permalink]

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New post 05 Jan 2015, 06:56
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Tough and Tricky questions: Geometry.



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If in the figure above, semicircles X and Y have radii of 6 and 8 feet, respectively, then what is the area of semicircle Z (in square feet)? (Note: Figure not drawn to scale.)

A. 10π
B. 20π
C. 25π
D. 50π
E. 100π


Kudos for a correct solution.

Attachment:
2015-01-05_1756.png
2015-01-05_1756.png [ 3.62 KiB | Viewed 1795 times ]

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Re: If in the figure above, semicircles X and Y have radii of 6 and 8 feet  [#permalink]

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New post 05 Jan 2015, 08:51
The pythagorean theorem is valid for squares as well as for circles, so the area of the semicircle z is equal to the sum of the semicircles x & y. Therefore:

\(\frac{6^2 \pi}{2} + \frac{8^2 \pi}{2} = 50 \pi\)

Answer D.
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Re: If in the figure above, semicircles X and Y have radii of 6 and 8 feet  [#permalink]

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New post 05 Jan 2015, 17:17
Bunuel wrote:

Tough and Tricky questions: Geometry.



Image
If in the figure above, semicircles X and Y have radii of 6 and 8 feet, respectively, then what is the area of semicircle Z (in square feet)? (Note: Figure not drawn to scale.)

A. 10π
B. 20π
C. 25π
D. 50π
E. 100π


Kudos for a correct solution.

Attachment:
2015-01-05_1756.png



diameter of of z = \sqrt{12^2 + 16^2}
Diameter of Z = 20; radius = 10

Area of Z = 10^2π / 2 =50π
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Re: If in the figure above, semicircles X and Y have radii of 6 and 8 feet  [#permalink]

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New post 06 Jan 2015, 00:27
Answer = D

Radius of circle Z\(= \sqrt{6^2 + 8^2} = 10\)

Area of the semicircle \(= \frac{\pi10^2}{2} = 50\pi\)
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Re: If in the figure above, semicircles X and Y have radii of 6 and 8 feet  [#permalink]

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New post 09 Jan 2015, 07:23
Bunuel wrote:

Tough and Tricky questions: Geometry.



Image
If in the figure above, semicircles X and Y have radii of 6 and 8 feet, respectively, then what is the area of semicircle Z (in square feet)? (Note: Figure not drawn to scale.)

A. 10π
B. 20π
C. 25π
D. 50π
E. 100π


Kudos for a correct solution.

Attachment:
2015-01-05_1756.png


OFFICIAL SOLUTION:

(D) If the radii of the semicircles X and Y are 6 feet and 8 feet, then their diameters are 12 and 16 feet. These diameters are both lengths of the legs of a right triangle, and the diameter of semicircle Z is the hypotenuse of this triangle.

The triangle is a 3-4-5 triangle, since the perpendicular legs are in the ratio 3 : 4. So the hypotenuse measures 20 feet, meaning the diameter of Z is 20 feet and its radius is 10 feet.

To determine the area of semicircle Z, we will determine the area of a full circle with the same radius as Z, and then divide this quantity by 2.The area of a circle with radius 10 is:
Area = πr²
Area = π10² = 100π.

Thus, the area of semicircle Z is 50π square feet.

The correct answer is choice (D).
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Re: If in the figure above, semicircles X and Y have radii of 6 and 8 feet   [#permalink] 09 Jan 2015, 07:23
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