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Great, Mike!

Thank you a lot for this clarification.

Honestly after so many wrong assumptions I have difficulty to trust any diagrams in DS :)

Cheers!
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We are given a right triangle that is cut into four smaller right triangles. Each smaller triangle was formed by drawing a perpendicular from the right angle of a larger triangle to that larger triangle's hypotenuse. When a right triangle is divided in this way, two similar triangles are created. And each one of these smaller similar triangles is also similar to the larger triangle from which it was formed.
Thus, for example, triangle ABD is similar to triangle BDC, and both of these are similar to triangle ABC. Moreover, triangle BDE is similar to triangle DEC, and each of these is similar to triangle BDC, from which they were formed. If BDE is similar to BDC and BDC is similar to ABD, then BDE must be similar to ABD as well.
Remember that similar triangles have the same interior angles and the ratio of their side lengths are the same. So the ratio of the side lengths of BDE must be the same as the ratio of the side lengths of ABD. We are given the hypotenuse of BDE, which is also a leg of triangle ABD. If we had even one more side of BDE, we would be able to find the side lengths of BDE and thus know the ratios, which we could use to determine the sides of ABD.
(1) SUFFICIENT: If BE = 3, then BDE is a 3-4-5 right triangle. BDE and ABD are similar triangles, as discussed above, so their side measurements have the same proportion. Knowing the three side measurements of BDE and one of the side measurements of ABD is enough to allow us to calculate AB.

To illustrate:
BD = 5 is the hypotenuse of BDE, while AB is the hypotenuse of ABD.
The longer leg of right triangle BDE is DE = 4, and the corresponding leg in ABD is BD = 5.

Since they are similar triangles, the ratio of the longer leg to the hypotenuse should be the same in both BDE and ABD.
For BDE, the ratio of the longer leg to the hypotenuse = 4/5.
For ABD, the ratio of the longer leg to the hypotenuse = 5/AB.
Thus, 4/5 = 5/AB, or AB = 25/4 = 6.25

(2) SUFFICIENT: If DE = 4, then BDE is a 3-4-5 right triangle. This statement provides identical information to that given in statement (1) and is sufficient for the reasons given above.

The correct answer is D.
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Hi,
I have a doubt in this question and similar questions like these. In this particular question I got the answer 25/3 because the proportion I set up was BE/BD=BD/AB. How does one identify which base and height of one 90 degree triangle matches with which height and base for another? Please help. It seems like I’m missing some obvious point. Thanks!

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geetikavig
Hi,
I have a doubt in this question and similar questions like these. In this particular question I got the answer 25/3 because the proportion I set up was BE/BD=BD/AB. How does one identify which base and height of one 90 degree triangle matches with which height and base for another? Please help. It seems like I’m missing some obvious point. Thanks!

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I think this is addressed couple of time in this thread. The main idea is that the ratio of the corresponding sides are equal (corresponding sides are the sides opposite the same angles).
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The Best approach shall be using graphical method, mentally drawing the figure to find whether the figure can be uniquely constructed with the information given.
if we can construct the figure uniquely, We can find all angles, length of sides , any data related to the figure.

See the graphical approach as per the Sketch attached.
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Bunuel My question here is if Triangle BDE and Triangle DEC are similar then their ratios of sides are equal hence BE/DE=EC/DE ====>3/4=EC/4, hence EC=3 also DC=5.
Now considering BDC a right triangle BD^2+DC^2=BC^2====> 5^2+5^2=6^2 ( which this equation is not justified).
Please correct where I am mistaken!!
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Hi

Triangle BDE is NOT similar to triangle DEC
Triangle BDE is similar to triangle DCE

Please now see, whether you have any doubt.

varun325
Bunuel My question here is if Triangle BDE and Triangle DEC are similar then their ratios of sides are equal hence BE/DE=EC/DE ====>3/4=EC/4, hence EC=3 also DC=5.
Now considering BDC a right triangle BD^2+DC^2=BC^2====> 5^2+5^2=6^2 ( which this equation is not justified).
Please correct where I am mistaken!!
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Hello from the GMAT Club BumpBot!

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