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The diameter of the circle is 10

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The diameter of the circle is 10 [#permalink] New post 27 Dec 2016, 03:17
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The diameter of the circle is 10

Quantity A
Quantity B
The area of the region enclosed by quadrilateral ABCD
40


A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.
[Reveal] Spoiler: OA

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Re: The diameter of the circle is 10 [#permalink] New post 28 Dec 2016, 16:39
Expert's post
Carcass wrote:
Image

The diameter of the circle is 10

Quantity A
Quantity B
The area of the region enclosed by quadrilateral ABCD
40



First of all, it's important not to read too much into the diagram.
All we can glean from the diagram is that we have a quadrilateral that is inscribed in the circle.
That's it!

So, first recognize that the inscribed quadrilateral COULD be a very very very narrow rectangle like this.
Image
Notice that this quadrilateral COULD be so thin that its area is very very close to zero.
So, for this particular quadrilateral we get:
Quantity A: a very very small area that's close to zero
Quantity B: 40
In this case, Quantity B is greater.


Alternatively, we COULD make the quadrilateral quite large.
In fact, we could make it a SQUARE.
Image

So, if the inscribed quadrilateral is a square, what is its area?
To find out, let's draw a diagonal.
Image
One of our circle properties tells us that this diagonal must be the diameter of the circle, which we know is 10
To find the area of the square, we need to know the length of each side.
So, let's let x = the length of each side.

Since ACD is a RIGHT TRIANGLE, we can apply the Pythagorean Theorem to get: x² + x² = 10²
Simplify to get 2x² = 100
Divide both sides by 2 to get: x² = 50

This means the area of the square = 50
We know this because the area of the square = (x)(x) = x², and we just learned that x² = 50

So, for this particular quadrilateral we get:
Quantity A: 50
Quantity B: 40
In this case, Quantity A is greater.

Answer: D

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Re: The diameter of the circle is 10 [#permalink] New post 28 Jul 2017, 10:56
Could you please explain why we need to consider that the inscribe quadrilateral has a changing size. I mean that can be consider as a small rectangle or a big square.
Thanks so much.
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Re: The diameter of the circle is 10 [#permalink] New post 08 Aug 2017, 01:45
But, if we know that the diameter of the circle is 10, then the area of the circle is 25p, is not it? Therefore, we can assume that anything that is inscribed in the circle is going to be lower then 25p.
Re: The diameter of the circle is 10   [#permalink] 08 Aug 2017, 01:45
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