If points A, B, C, and D are points on the circumference of the circle

Question

https://gmatclub.com/forum/download/file.php?id=45503 If points A, B, C, and D are points on the circumference of the circle in the figure above, what is the area of ABCD? (1) The radius of the circle is \frac{\sqrt{2}}{2} . (2) ABCD is a square. 2018-12-12_1448.png

Archit3110 · Answer

from 1 :
we can determine the diameter of the circle which is \sqrt{2}

so side AD and BC = 1/sqrt2

but its not know whether ABCD is a square or rectangle ; insufficient

From 2 given is ABCD is a square but sides value is not know so in sufficient

using 1 &2 we can find are of ABCD 
OPTION C

Ypsychotic · Answer

1) Since only radius is given, nothing can be deduced about the right angles it maintains at the circle. Not Sufficient.

2) ABCD is a sqaure. Also, a square is type of a rhombus. Area of a rhombus =1/2*d1*d2. Since, Square is also a type of rectangle. d1=d2. Area pf square = 1/2*d*d, where d is the diameter. But we don't what is the diameter of cirlce.

1+2, Area= 1/2*d*d, and d=2r=2* \frac{\sqrt{2}}{2}

So, Area= 1, Sufficient. IMO, Option C.

Archit3110 · Answer

GMATinsight  :

Sir could you please review on why is my assumption of stmnt 1 wrong? cant it be sufficient because given in figure is 90 * and it is a diagonal , we can split figure into two 
triangles and solve for area ?

750Barrier · Answer

only option 1 is not sufficient... because u never know that is a square or a rectangle.

srishti246 · Answer

This is right. Experts please confirm. Using the property 90 degrees inscribed triangle divides the circle into two equal halves, why isn't statement A enough?

randomavoidplease · Answer

Hi same question, why is it not enough? I am not looking at it on a square method, I am looking at it as an iscolesces triangle. Where the hypotenuse is root2. Thus b = h= 1. Thus we have area and we can multiply by 2 to get area of both sides.

Seems fine to me, why is that not the case?

Archit3110 · Answer

Seeing figure you cannot assume that it would be square ..
Which is why option A is incorrect

Posted from my mobile device

randomavoidplease · Answer

But I am not assuming it is a square, I am saying it is a right triangle with 45 45 90 angles as the hypotenuse is  \sqrt{2} . Thus the ratio is  x:x:x\sqrt{2} .

andylapian · Answer

Hi, we cannot assume the triangle inside the the circle to be isosceles triangle. There is a possibility that the triangle inside is 30-60-90 degree triangle, or even 20-70-90 triangle. We just know that one of the angles is 90 degree and that is all. Thus, statement (1) is not enough.

Statement (2) state our assumption that the triangle is 45-45-90 degree triangle, thus we do not know the length.

Thus we need (1) + (2) to solve this problem.

Statement (2) state our assumption that the triangle is 45-45-90 degree triangle, thus we do not know the length.

Thus we need (1) + (2) to solve this problem.

Archit3110 · Answer

45:45:90 is only possible when two sides are same hence a square isn't it.. Likewise it can also be 30:60:90

but it isn't

srishti246 · Answer

But if you draw a perpendicular from DO to AC, the radius sides or DO and OC become equal, making angle ACD 45 degrees. Please correct me if I am wrong.

750Barrier · Answer

please consider and remember -- we can inscribe a rectangle in a circle and we can inscribe a square in a circle.. -- both the diag. of rectangle are equal, just as the diag of square are.. now considering above 2 points, you will understand that A alone is not sufficient to confirm its a square..

---------------- Sir could you please review on why is my assumption of stmnt 1 wrong? cant it be sufficient because given in figure is 90 * and it is a diagonal , we can split figure into two triangles and solve for area ? even diag of rect will divide the rect into 2 halves.

kntombat · Answer

Can  GMATinsight  ,  Bunuel  ,  chetan2u  please help us out ?

kush09 · Answer

Here the assumption that angle DOC =90 degree is incorrect, it could be anything from 60 to higher value, please refer below diagram,

chetan2u · Answer

kntombat

Please look at the sketch attached.

As it is given that B and D are 90, AC MUST be the diameter. BUT B and D can be anywhere on the circumference, so ABCD can be a square, rectangle or any quadrilateral with opposite angles 90.

(1) The radius of the circle is  \frac{\sqrt{2}}{2} .
As we do not know the exact place of B and D, we cannot calculate the area.

(2) ABCD is a square.
We do not know the radius

combined
we can find the area as we know the diagonal AC, and can find the sides.
Suff

C

If points A, B, C, and D are points on the circumference of the circle

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If points A, B, C, and D are points on the circumference of the circle

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