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by srcc25anu » Mon Mar 28, 2011 5:41 am
Quadrilateral ABCD is inscribed in circle K. The diameter of K is 20. AC is perpendicular to BD. What is the area of ABCD?

(1) AB = AD

(2) The length of CE is 8.
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by Anurag@Gurome » Mon Mar 28, 2011 8:05 am
srcc25anu wrote:Quadrilateral ABCD is inscribed in circle K. The diameter of K is 20. AC is perpendicular to BD. What is the area of ABCD?

(1) AB = AD

(2) The length of CE is 8.
Image
AC is the diameter of the circle. So, AC = 20
Triangles CBA and CDA are congruent. So, area of ABCD = 2 * area of triangle CDA = 2 * 1/2 * DE * AC = 20DE. Now we want the value of DE.

(1) AB = AD does not give the value of DE. So, (1) is NOT SUFFICIENT.

(2) CE = 8 and AC = 20. It can noticed that CE + EA = AC implies 8 + EA = 20 or EA = 12
Also, triangles DAE and CDE are similar triangles. So, corresponding sides will be in equal proportion. Hence, DE/EA = CE/DE or DE² = EA*CE.
So, DE² = 12*8 = 96, which implies we can find DE and hence area of ABCD. So, (2) is SUFFICIENT.

The correct answer is B.
Last edited by Anurag@Gurome on Mon Mar 28, 2011 7:00 pm, edited 1 time in total.
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by clock60 » Mon Mar 28, 2011 12:21 pm
hi Anurag
can you explain a little bit more how you came that AC and BD are the diameters of the circle
thanks
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by srcc25anu » Mon Mar 28, 2011 2:15 pm
same doubt as clock60: its not given that BD is diameter??? AC = diameter (diagram suggests so)
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by Anurag@Gurome » Mon Mar 28, 2011 7:04 pm
srcc25anu wrote:same doubt as clock60: its not given that BD is diameter??? AC = diameter (diagram suggests so)
You are right AC is the diameter, but BD is not the diameter. Somehow, I mistakenly wrote that and didn't used that anywhere in the solution. I have edited the previous post. I hope it's clear now?
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by srcc25anu » Mon Mar 28, 2011 7:45 pm
Thanks Anurag, got it.
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by Anurag@Gurome » Mon Mar 28, 2011 7:46 pm
srcc25anu wrote:Thanks Anurag, got it.
You are welcome.
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by manpsingh87 » Mon Mar 28, 2011 7:51 pm
srcc25anu wrote:Quadrilateral ABCD is inscribed in circle K. The diameter of K is 20. AC is perpendicular to BD. What is the area of ABCD?

(1) AB = AD

(2) The length of CE is 8.
Image
Quadrilateral ABCD is a kite..!! whose diagonals are intersecting at 90 degree, and area of kite is equal to product of its diagonals, we know one of the diagonals which is 20 as per the question, now lets just analyze the two statements.

1) AB=CD; we will not be able to find the length of the other diagonal of the kite by using this information hence 1 is not sufficient to answer the question.

2) CE=8, as we know that diameter ac of k is 20 therefore its radius is 10, i.e. CK=10, KE=CK-CE;

KE=10-8=2;

now draw draw the line connecting K to B, KB=10( radius of a circle);
as we know that in a circle, perpendicular drawn from the center of the circle bisects the chord, k is a center of the circle, and BD is chord, therefore BE=ED;

now in right angle triangle KEB; we have KB^2=BE^2+EK^2; KB=10,EK=2,BE=x;

100=x^2+4;
x=sqrt(96);

therefore BD= 2(BE)= 2sqrt(96);

therefore Area=1/2 AC*BD; 1/2 *20*2sqrt(96)= 20sqrt(96);

hence statement 2 alone is sufficient to answer the question hence B
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