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# M25-14

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Math Expert
Joined: 02 Sep 2009
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16 Sep 2014, 01:23
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Difficulty:

75% (hard)

Question Stats:

42% (01:04) correct 58% (01:08) wrong based on 321 sessions

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

(1) AC is perpendicular to BD

(2) $$AB + CD = BC + AD$$

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16 Sep 2014, 01:23
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Official Solution:

Based on geometry, $$ABCD$$ can be a rhombus ($$AB = BC = CD = DA$$) but it can also be a kite-shaped figure whose diagonals form a cross (S1 holds). S2 holds because of the symmetry of the kite ($$AB = BC$$ and $$CD = AD$$).

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06 Dec 2014, 03:03
but if both the diagonals are perpendicular to each other (Option-1) , then it could be RHOMBUS only. then its all the sides are equal.
Which other quadilateral have perpendicular diagonals? Square ! that means it could be square also and square also has all sides equal.

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06 Dec 2014, 06:13
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manojpandey80 wrote:
but if both the diagonals are perpendicular to each other (Option-1) , then it could be RHOMBUS only. then its all the sides are equal.
Which other quadilateral have perpendicular diagonals? Square ! that means it could be square also and square also has all sides equal.

Both kite and rhombus have the diagonals perpendicular to each other and the sum of the opposite sides equal to each other. So, ABCD can be a kite or a rhombus. Look at the diagram below:

Alternative solution:
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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30 Aug 2015, 01:59
I think this is a high-quality question and the explanation isn't clear enough, please elaborate. In a rhombus all four sides are equal.
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30 Aug 2015, 08:35
aganesh wrote:
I think this is a high-quality question and the explanation isn't clear enough, please elaborate. In a rhombus all four sides are equal.

Have you checked this post: m25-184416.html#p1451996 ?
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20 Sep 2015, 11:19
I think bunuel explanation is more than sufficient. Thanks. The diagrams helped. I had trouble imagining the object, but then again I suck at geometry.
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26 Jul 2016, 11:10
I think this is a high-quality question and I agree with explanation.
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16 Dec 2017, 08:01
manojpandey80 wrote:
but if both the diagonals are perpendicular to each other (Option-1) , then it could be RHOMBUS only. then its all the sides are equal.
Which other quadilateral have perpendicular diagonals? Square ! that means it could be square also and square also has all sides equal.

Diagonals are perpendicular in parallelogram, rectangle, square and rhombus.
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16 Dec 2017, 09:18
Ashokshiva wrote:
manojpandey80 wrote:
but if both the diagonals are perpendicular to each other (Option-1) , then it could be RHOMBUS only. then its all the sides are equal.
Which other quadilateral have perpendicular diagonals? Square ! that means it could be square also and square also has all sides equal.

Diagonals are perpendicular in parallelogram, rectangle, square and rhombus.

The diagonal in parallelogram and rectangle are NOT perpendicular to each other.

For more practice Properties of Polygons Questions.
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17 Dec 2017, 10:32
Bunuel wrote:
Ashokshiva wrote:
manojpandey80 wrote:
but if both the diagonals are perpendicular to each other (Option-1) , then it could be RHOMBUS only. then its all the sides are equal.
Which other quadilateral have perpendicular diagonals? Square ! that means it could be square also and square also has all sides equal.

Diagonals are perpendicular in parallelogram, rectangle, square and rhombus.

The diagonal in parallelogram and rectangle are NOT perpendicular to each other.

For more practice Properties of Polygons Questions.

thanks

The diagonals in parallelogram and rectangle only bisect each other and are not perpendicular.
Re: M25-14   [#permalink] 17 Dec 2017, 10:32
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# M25-14

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