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Points A, B, C, and D form a quadrilateral. Is AC longer

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Points A, B, C, and D form a quadrilateral. Is AC longer [#permalink]

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New post 09 May 2009, 16:40
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A
B
C
D
E

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Points A, B, C, and D form a quadrilateral. Is AC longer than BD?

1. angle ABC > angle BCD
2. AB = BC = CD = DA

* Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient
* Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient
* BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
* EACH statement ALONE is sufficient
* Statements (1) and (2) TOGETHER are NOT sufficient

Kudos [?]: 437 [0], given: 1

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Kudos [?]: 101 [0], given: 10

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Re: Quadrilateral and its diagonals [#permalink]

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New post 09 May 2009, 18:04
Points A, B, C, and D form a quadrilateral. Is AC longer than BD?

1. angle ABC > angle BCD
2. AB = BC = CD = DA


Stat.1
angle ABC > angle BCD
not sufficient.

Stat.2
If AB = BC = CD = DA, then it is either a square or rhombus.Hence AC=BD.


Answer=B.
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Kudos [?]: 101 [0], given: 10

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Re: Quadrilateral and its diagonals [#permalink]

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New post 11 May 2009, 04:04
The answer is C.

Statement 1 just tells us that two adjacent internal angles are not identical within a quadrilateral.
Hence Statement 1 alone is INSUFFICIENT

Statement 2 alone tells you that it is a rhombus (a square is just a special case of a rhombus)
For a rhombus, the two diagonals are not of the same length unless it happens to also be a square.
Hence Statement 2 alone is INSUFFICIENT.

If you combine Statement 1 and 2 then you have a SUFFICIENT condition.
You know it's a rhombus, so the sides are equal length, and you know one angle is larger than the other.
Therefore you know that AC is the longer diagonal of the two. If this not obvious to you, draw it out on a piece of paper, and it'll immediately come to you.

If you want the mathematical explanation [WHICH IS NOT NECESSARY FOR DATA SUFFICIENCY], then you need to know the cosine rule.

Using the cosine rule, we know that
|AC|^2 = |AB|^2 + |BC|^2 - 2|AB||BC|cos(angle ABC)
|BD|^2 = |BC|^2 + |CD|^2 - 2|BC||CD|cos(angle BCD)

Since it is a rhombus, |AC| = |BC| = |CD|, and we'll call this length x.
So |AC|^2 - |BD|^2 = 2(x^2)(cos(angle BCD) - cos(angle ABC))

angle (BCD) < angle (ABC), therefore cos(angle BCD) > cos(angle ABC)

Hence |AC|^2 - |BD|^2 > 0, so |AC| > |BD|

Kudos [?]: 29 [0], given: 6

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Joined: 16 Apr 2009
Posts: 231

Kudos [?]: 101 [0], given: 10

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Re: Quadrilateral and its diagonals [#permalink]

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New post 11 May 2009, 10:46
"I think the problem with B is that a rhombus may have angles other than 90 degrees (don't quote me on this though), but still have sides that are equal in length. If the angles are not all equal to 90 degrees, than AC must not equal to BD. (B) alone does not tell us the angle properties of this rhombus/square/quadrilateral, we can only find this out from statement 1."


I agree with you,Typhoid X.I need to hone my skills.
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Keep trying no matter how hard it seems, it will get easier.

Kudos [?]: 101 [0], given: 10

Re: Quadrilateral and its diagonals   [#permalink] 11 May 2009, 10:46
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Points A, B, C, and D form a quadrilateral. Is AC longer

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