Is quadrilateral ABCD a square? (1) A = C = 90º (2) AB = AD

Question

Is quadrilateral ABCD a square?

(1) A = C = 90º

(2) AB = AD

Bunuel · Answer

Check other  Properties of Polygons Questions  from our  Special Questions Directory .

chetan2u · Answer

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
AB=AD
nothing much 
insuff

combined..

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

E

NandishSS · Answer

chetan2u  Bhai,

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

amanvermagmat · Answer

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.

Bunuel · Answer

There is a smart way to prove that ABCD may or may not be a square. https://gmatclub.com/forum/download/file.php?id=40010 Case 1: consider AB C D to be a square inscribed in a circle. A = C = 90º, AB = AD and AB C D 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 B C D will also be a right angle. A = C = 90º, AB = AD and AB C D is NOT a square. Answer: E. Untitled.png

NandishSS · Answer

amanvermagmat , chetan2u , Bunuel , Thanks for Solution, Will take it a learning

Happy Prepping

AliciaSierra · Answer

Bunuel 
Amazing. You made this qoestion so easy.

Zyanpandit25 · Answer

the diagonal should bisect each other for the quad to be a square apart from each angle being 90

bumpbot · Answer

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Is quadrilateral ABCD a square? (1) A = C = 90º (2) AB = AD

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