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# m25 q14

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Intern
Joined: 17 Mar 2010
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13 May 2010, 12:29
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If ABCD is a quadrilateral, is AB = BC = CD = DA?

1. AC is perpendicular to BD
2. $$AB + CD = BC + AD$$

Answer is E. But as far as I learned in school, the ONLY 2 quadrilaterals that have perpendicular diagonals are the rhombus and the square and both have all sides equal. I do not find the explanation correct.
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Manager
Joined: 18 Mar 2010
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13 May 2010, 13:00
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Not sure if my reasoning is correct, but this is what I got.

Draw a kite shape, with A being the top tip of the kite, and C being the bottom tip; B and D are opposite side corners. AC and BD (where the kite frame sticks would go) are perpendicular. And since all perfect kites are symmetric, then the sides AB=AD, and BC=CD, so AB+CD=BC+AD. But obviously AB does not equal BC. Or you would have a square kite. Answer is E.

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Manager
Joined: 19 Oct 2011
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18 Jul 2012, 18:36
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Could you please explain statement wise why they are insufficient?
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Math Expert
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19 Jul 2012, 03:17
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dvinoth86 wrote:
Bunnel

Could you please explain statement wise why they are insufficient?

If ABCD is a quadrilateral, is AB=BC=CD=DA ?

(1) AC is perpendicular to BD. The diagonals are perpendicular to each other: ABCD could be a kite (answer NO), a rhombus (answer YES) or a square, which is just a special type of rhombus (answer YES). Not sufficient.

(2) AB+CD=BC+DA. The sum of opposite sides are equal. Clearly insufficient.

(1)+(2) ABCD could be a kite (see the diagram below) - answer NO or a square/rhombus - answer YES. Not sufficient.

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Re: m25 q14   [#permalink] 19 Jul 2012, 03:17
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# m25 q14

Moderator: Bunuel

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