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Math Expert V
Joined: 02 Sep 2009
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Is quadrilateral ABCD a square? (1) A = C = 90º (2) AB = AD  [#permalink]

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10 00:00

Difficulty:   75% (hard)

Question Stats: 32% (01:02) correct 68% (00:54) wrong based on 153 sessions

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Is quadrilateral ABCD a square?

(1) A = C = 90º

(2) AB = AD

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Math Expert V
Joined: 02 Sep 2009
Posts: 58453
Is quadrilateral ABCD a square? (1) A = C = 90º (2) AB = AD  [#permalink]

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9
4
Bunuel wrote:
Is quadrilateral ABCD a square?

(1) A = C = 90º

(2) AB = AD

There is a smart way to prove that ABCD may or may not be a square. Case 1: consider ABCD to be a square inscribed in a circle.

A = C = 90º, AB = AD and ABCD is a square.

Case 2: consider upper right semicircle BCD, with diameter BD. If the diameter of a circle is also the triangle’s side, then that triangle is a right triangle (the reverse is also true: a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle.) According to this property ANY point C on this semicircle will create a right angle BCD. For example, angle BCD will also be a right angle.

A = C = 90º, AB = AD and ABCD is NOT a square.

Attachment: Untitled.png [ 14.65 KiB | Viewed 2359 times ]

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Math Expert V
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Posts: 58453
Re: Is quadrilateral ABCD a square? (1) A = C = 90º (2) AB = AD  [#permalink]

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Bunuel wrote:
Is quadrilateral ABCD a square?

(1) A = C = 90º

(2) AB = AD

Check other Properties of Polygons Questions from our Special Questions Directory.
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Posts: 7978
Re: Is quadrilateral ABCD a square? (1) A = C = 90º (2) AB = AD  [#permalink]

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1
Bunuel wrote:
Is quadrilateral ABCD a square?

(1) A = C = 90º

(2) AB = AD

Ans will not be C but E..

statement I..
A=C=90

the triangle can be rectangle or square or any other possiblity too..
Insuff

statement II
nothing much
insuff

combined..

you can still have it as any other quadrilateral or as a square
see attached figure

E
Attachments rectangles.png [ 7.11 KiB | Viewed 2287 times ]

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Re: Is quadrilateral ABCD a square? (1) A = C = 90º (2) AB = AD  [#permalink]

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chetan2u wrote:
Bunuel wrote:
Is quadrilateral ABCD a square?

(1) A = C = 90º

(2) AB = AD

Ans will not be C but E..

statement I..
A=C=90

the triangle can be rectangle or square or any other possiblity too..
Insuff

statement II
nothing much
insuff

combined..

you can still have it as any other quadrilateral or as a square
see attached figure

E

chetan2u Bhai,

How could be A = C = 90º in the first fig?
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Re: Is quadrilateral ABCD a square? (1) A = C = 90º (2) AB = AD  [#permalink]

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2
NandishSS wrote:
chetan2u wrote:
Bunuel wrote:
Is quadrilateral ABCD a square?

(1) A = C = 90º

(2) AB = AD

Ans will not be C but E..

statement I..
A=C=90

the triangle can be rectangle or square or any other possiblity too..
Insuff

statement II
nothing much
insuff

combined..

you can still have it as any other quadrilateral or as a square
see attached figure

E

chetan2u Bhai,

How could be A = C = 90º in the first fig?

Hi Nandish

I think it IS indeed possible for a quadrilateral ABCD to have AB=AD, angle A = angle C = 90 degrees, and yet not be a square.

I have attached a pic of a figure similar to that of Chetan's first figure. If you would look at my pic, I first made AB=AD = 3 cm each, with an angle of 90 degrees between them. Then I made a tilted long line DX from D (this line is obviously NOT inclined at 90 with AD) - then on this line DX I chose one particular point C, from where CB will be aligned at 90 degrees to DX.

In my figure, DC is < 3 and CB is > 3. So its a figure slightly similar to Chetan's figure, where both conditions of question are being satisfied, yet its not a square.
Attachments IMG_20180214_203201_3.jpg [ 23.12 KiB | Viewed 2224 times ]

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Re: Is quadrilateral ABCD a square? (1) A = C = 90º (2) AB = AD  [#permalink]

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amanvermagmat, chetan2u, Bunuel,

Thanks for Solution, Will take it a learning Happy Prepping _________________
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Is quadrilateral ABCD a square? (1) A = C = 90º (2) AB = AD  [#permalink]

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Bunuel wrote:
Bunuel wrote:
Is quadrilateral ABCD a square?

(1) A = C = 90º

(2) AB = AD

There is a smart way to prove that ABCD may or may not be a square. Case 1: consider ABCD to be a square inscribed in a circle.

A = C = 90º, AB = AD and ABCD is a square.

Case 2: consider upper right semicircle BCD, with diameter BD. If the diameter of a circle is also the triangle’s side, then that triangle is a right triangle (the reverse is also true: a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle.) According to this property ANY point C on this semicircle will create a right angle BCD. For example, angle BCD will also be a right angle.

A = C = 90º, AB = AD and ABCD is NOT a square.

Attachment:
Untitled.png

Bunuel
Amazing. You made this qoestion so easy. Is quadrilateral ABCD a square? (1) A = C = 90º (2) AB = AD   [#permalink] 28 Aug 2018, 08:41
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