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# M20-24

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Math Expert
Joined: 02 Sep 2009
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16 Sep 2014, 01:09
Expert's post
5
This post was
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00:00

Difficulty:

45% (medium)

Question Stats:

64% (00:39) correct 36% (00:54) wrong based on 74 sessions

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What is the area of rectangle $$ABCD$$ ?

(1) Diagonal $$AC$$ is twice as long as side $$CD$$

(2) Diagonal $$AC$$ is 0.1 meters longer than side $$AD$$
[Reveal] Spoiler: OA

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Kudos [?]: 128857 [0], given: 12183

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16 Sep 2014, 01:09
Expert's post
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Official Solution:

Statement (1) by itself is insufficient. S1 defines the shape of the rectangle but does not specify its size.

Statement (2) by itself is insufficient. The area of the rectangle can be made arbitrarily small (as the length of side $$AD$$ goes to 0).

Statements (1) and (2) combined are sufficient. Because $$AC^2 = CD^2 + AD^2$$ and $$AC = 2CD$$, $$4CD^2 = CD^2 + AD^2$$ which simplifies to $$3CD^2 = AD^2$$. It follows from here that $$AD = \frac{\sqrt{3}}{2} AC$$.

This is one of the equations, the other one is $$AC = AD + 0.1$$. This system of linear equations is enough to determine both sides of the rectangle.

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26 Jul 2016, 12:21
Hi Bunuel,

Wouldn't (1) work with the triangle type x:xsqrt3:2x?

Thanks

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26 Jul 2016, 12:39
dauren88 wrote:
Hi Bunuel,

Wouldn't (1) work with the triangle type x:xsqrt3:2x?

Thanks

It will
But area of the said rectangle will change I.e directly proportional to x
Hence in suff

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16 Aug 2017, 07:04
Statements (1) and (2) combined are sufficient. Because $$AC^2 = CD^2 + AD^2$$ and $$AC = 2CD$$, $$4CD^2 = CD^2 + AD^2$$ which simplifies to $$3CD^2 = AD^2$$. [color=#fff200]It follows from here that $$AD = \frac{\sqrt{3}}{2} AC$$.[/color]

This is one of the equations, the other one is $$AC = AD + 0.1$$. This system of linear equations is enough to determine both sides of the rectangle.

Hello Bunuel,
Can you please let me know how have you arrived at the highlighted text?

Thanks!

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16 Aug 2017, 07:12
anirudhasopa wrote:
Statements (1) and (2) combined are sufficient. Because $$AC^2 = CD^2 + AD^2$$ and $$AC = 2CD$$, $$4CD^2 = CD^2 + AD^2$$ which simplifies to $$3CD^2 = AD^2$$. [color=#fff200]It follows from here that $$AD = \frac{\sqrt{3}}{2} AC$$.[/color]

This is one of the equations, the other one is $$AC = AD + 0.1$$. This system of linear equations is enough to determine both sides of the rectangle.

Hello Bunuel,
Can you please let me know how have you arrived at the highlighted text?

Thanks!

$$3CD^2 = AD^2$$

Take the square root from both sides: $$\sqrt{3}CD = AD$$.

We know that $$AC = 2CD$$, so $$\frac{AC}{2} = CD$$.

Thus, $$\sqrt{3}*\frac{AC}{2} = AD$$.

Hope it's clear.
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16 Oct 2017, 06:09
+1 for option C.
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Re: M20-24   [#permalink] 16 Oct 2017, 06:09
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# M20-24

Moderators: Bunuel, Vyshak

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