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mikemcgarry
Attachment:
quadrilateral inside circles.png
In the diagram above, the four triangles ABE, CBE, ADE, and CDE are all equal, and CD = 5. What is the area between the two circles?

Statement #1: AE = 3

Statement #2: angle BEC = 90 degrees


Geometry is beautiful, and the GMAT DS loves geometry. This problem is a from a collection of ten practice problems. For the other problems, as well as the OE for this question, see:
GMAT Data Sufficiency Geometry Practice Questions

Mike :-)

Each of the 4 equal triangles is such that the bigger radius forms one the sides, the smaller radius forms the other side and the length of hypotenuse is given (CD = 5).

Statement 1

AE =3, AB = CD = 5, Therefore BE is 4 (3, 4, 5 is a Pythagorean pair. Triangle ABE is a right angled \(/triangle\))

The bigger radius is known. Area of bigger circle is known.
Smaller radius is known. Area of smaller circle is known.
We can calculate the area of the space between two circles

Sufficient.

Statement 2
This information is already known and does not give any additional information. Further, we cannot calculate the radius of the two circles.

Hence not sufficient

A is the answer.
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robu
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mikemcgarry
Attachment:
quadrilateral inside circles.png
In the diagram above, the four triangles ABE, CBE, ADE, and CDE are all equal, and CD = 5. What is the area between the two circles?

Statement #1: AE = 3

Statement #2: angle BEC = 90 degrees


Geometry is beautiful, and the GMAT DS loves geometry. This problem is a from a collection of ten practice problems. For the other problems, as well as the OE for this question, see:
GMAT Data Sufficiency Geometry Practice Questions

Mike :-)

Hi,

If you look at ABE and CBE, they are congruent triangles....
HOW?-
since B and D are at equal distance from A and C, the line drawn through B and D will also be at equal distance at every point so E will be at SAME distance from A and C..
so AE = CE...
all sides are equal so triangles are congruent..


therefore all angles will be equal and angle AEB= angle CEB = 90..

lets see the statements now..



Statement #1: AE = 3
Now we know AE and AB in right angle triangle ABE...
so \(BE^2 = AB^2-AE^2=5^2-3^2 = 4^2..................BE=4\)

radius of both circles is known we can find the differenjce in AREA..
Suff

Statement #2: angle BEC = 90 degrees
we have already derived this info
Insuff






In question stem it is given: " the four triangles ABE, CBE, ADE, and CDE are all equal". Triangles equal does not mean that they are equilateral. , Their areas are equal.

Please clarify more .

Hi robu ,
Triangles are all equal means they are congruent .
If two or more triangles are congruent, then all of their corresponding angles and sides are congruent as well.
Here Since AE= 3 , EC= 3
Also, AB = BC = CD = AD = 5
and \(\angle\) BEA = \(\angle\) BEC = \(\angle\) CED = \(\angle\) DEA = 90

Hope this helps!! :)
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If the angle is 90 degree in statement 2. Isnt it obvious that the other two sides would be 3 & 4 as the 3,4,5 are pythagorean triplets. Bunuel
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007saisurya


In the diagram above, the four triangles ABE, CBE, ADE, and CDE are all equal, and CD = 5. What is the area between the two circles?

Statement #1: AE = 3

Statement #2: angle BEC = 90 degrees


If the angle is 90 degree in statement 2. Isnt it obvious that the other two sides would be 3 & 4 as the 3,4,5 are pythagorean triplets. Bunuel

Two points:

1. It can be inferred that angle BEC equals 90 degrees from the stem of the problem itself. In triangles ABE, CBE, ADE, and CDE, all angles are equal, and since the sum of the angles' measures at vertex E is 360 degrees, each angle at E must be 90 degrees. Therefore, when you assert that statement (2) is sufficient, you are actually suggesting that the problem's stem itself provides enough information to answer the question, implying that the question is flawed.


2. Just because CD, the hypotenuse in triangle CDE, has a length of 5, it doesn't necessarily mean that the sides of CDE form a Pythagorean triplet. We are not given that the side lengths must be integers. In other words, EC^2 + ED^2 = 5^2 has infinitely many solutions for EC and ED, with only one of them being EC = 3 and ED = 4. For instance, consider EC = 1 and ED = √24; EC = 2 and ED = √21; EC = 1/2 and ED = √24.75, and so on. If we were given that the lengths of EC and ED are integers, then a hypotenuse of 5 would imply that EC and ED are 3 and 4. Otherwise, knowing the length of the hypotenuse alone is insufficient to determine the other two sides.

Hope it's clear.
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