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what is the area of semicircle Z (in square feet)?

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by AAPL » Sat Dec 30, 2017 7:45 am
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In the figure above, semicircles X and Y have radii of 6 and 8 feet, respectively, then what is the area of semicrcle Z (in square feet)? (Note: figure not drawn to scale)

A. 10Ï€
B. 20Ï€
C. 25Ï€
D. 50Ï€
E. 100Ï€

The OA is D.

With the radii of semicircles X and Y I can get the diameter of semicirlce Z (using Pythagoras Theorem) and then I can get it area, right?

I appreciate if any expert explain it for me this PS question. Thank you so much.
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by [email protected] » Sat Dec 30, 2017 11:20 am
Hi AAPL,

We're told that semicircles X and Y have radii of 6 and 8 feet, respectively. We're asked for the area of semicrcle Z in square feet.

To start, since the radii of X and Y are 6 ft and 8 ft, their respective diameters are 12 ft and 16 ft.; these two measurements are the two 'legs' of the right triangle. At this point, you can use the Pythagorean Theorem to figure out the third side OR you can recognize that this is a 3/4/5 right triangle that has been 'quadrupled' (meaning the sides are 12, 16 and 20.

The diameter of Region Z is 20, so it's radius is 10. We have a semi-circle, so the area is 1/2 what the full circle would be....

Area of Z = (1/2)(pi)(R^2) = (1/2)(pi)(100) = 50pi

Final Answer: D

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what is the area of semicircle Z (in square feet)?

by EconomistGMATTutor » Sat Dec 30, 2017 11:41 am
In the figure above, semicircles X and Y have radii of 6 and 8 feet, respectively, then what is the area of semicrcle Z (in square feet)? (Note: figure not drawn to scale)

A. 10Ï€
B. 20Ï€
C. 25Ï€
D. 50Ï€
E. 100Ï€

The OA is D.

With the radii of semicircles X and Y I can get the diameter of semicirlce Z (using Pythagoras Theorem) and then I can get it area, right?

I appreciate if any expert explain it for me this PS question. Thank you so much.
Hi AAPL,
Let's take a look at your question.

Radius of the semi circle X = 6 ft
Diameter of the semi circle X = 12ft

Radius of the semi circle Y = 8 ft
Diameter of the semi circle Y = 16 ft

We can find the diameter of the semi circle Z using Pythagoras Theorem,
$$Diameter=\sqrt{12^2+16^2}$$
$$Diameter=\sqrt{144+256}$$
$$Diameter=\sqrt{400}=20$$

Radius of the semi circle Z = 20/2 = 10

$$\text{Area of Semi Circle Z}\ =\ \frac{1}{2}\pi r^2$$
$$\text{Area of Semi Circle Z}\ =\ \frac{1}{2}\pi (10)^2$$
$$\text{Area of Semi Circle Z}\ =\ \frac{1}{2}\pi (100)$$
$$\text{Area of Semi Circle Z}\ =\ 50\pi$$

Therefore, Option D is correct.

Hope it helps.
I am available if you'd like any follow up.
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