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In the diagram, what is the length of AB?

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Magoosh GMAT Instructor
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Re: In the diagram, what is the length of AB? [#permalink]

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New post 08 Dec 2015, 15:48
BrainLab wrote:
I have solved this one , but I have one question though... It was not really important (because its a DS) which sides to (short or long leg) to write in tha ratio, BUT for a PS question it's important. HOw one can know which sides are corresponding ? In this picture it's not easy to identify short and long leg....?

Dear BrainLab,
I'm happy to respond. :-) We can figure out everything we need about proportional sides by keeping track of the angles. Here's what I mean.

1) Look at the big triangle, ABC. Right angle at B. Say that the angle BAD has a measure of A and the angle DCE has a measure of C. We know that A and C are complementary angles, that is, A + C = 90 degrees. It appears that A > C, which means that BC > AB. Let's assume that is the relationship of these two angles.

2) Look at triangle ABD. Right angle at C. The angle at vertex A still has a measure of A, so the angle at vertex B, angle ABD, just have a measure of C. Again, A > C, so (angle BAD) > (angle ABD), which means that BD > AD.

3) Look at triangle DEC. Right angle at C. The angle at vertex C still has a measure of C, so the angle at vertex D, angle EDC, just have a measure of A. Again, A > C, so (angle EDC) > (angle ECD), which means that EC > DE.

4) Now, look at triangle BED. Right angle at E.
angle (BDE) = 90 - (angle EDC) = 90 - A = C
angle (DBE) = 90 - (angle ABD) = 90 - C = A
Again, A > C, so (angle DBE) > (angle BDE), which means that ED > BE.

Here is a highly accurate diagram of the situation, with the angles color-coded so you can see which angle equals which.
Attachment:
triangle with colored angles.JPG
triangle with colored angles.JPG [ 17.12 KiB | Viewed 852 times ]

Notice that the light green angle (A) plus the purple angle (C) equals a right angle, and that every triangle in the diagram has one light green angle, one purple angle, and one right angle. In each case, the leg opposite the purple angle is less than the leg opposite the light green angle.

Does all this make sense?
Mike :-)
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Re: In the diagram, what is the length of AB? [#permalink]

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New post 18 Jul 2016, 03:17
I disagree with this question.

It should be clearly stated that the line BD goes from the base to the vertex B. We can't simply trust the image and assume that the line goes straight to B. What if it is slightly on the left\right?

I think that the answer should be E, although I understand the "intention" of the question is to be D.
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Re: In the diagram, what is the length of AB? [#permalink]

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New post 18 Jul 2016, 08:02
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iliavko wrote:
I disagree with this question.

It should be clearly stated that the line BD goes from the base to the vertex B. We can't simply trust the image and assume that the line goes straight to B. What if it is slightly on the left\right?

I think that the answer should be E, although I understand the "intention" of the question is to be D.

Dear iliavko,
I'm happy to respond. :-)

My friend, there are a few things that we always can assume on GMAT diagram. One is that if a line segment appears to connect to a point, it actually does connect to that point. You yourself referred to that segment as BD: we don't even have a way of talking about it except to refer to its endpoints, so it would be duplicitous for the GMAT to allow for the segment to have an endpoint something other then the obvious endpoint. If any segment looks like it goes to a point, you can assume that it is God-given truth that the segment in fact does go to that point.

Does all this make sense?
Mike :-)
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Re: In the diagram, what is the length of AB? [#permalink]

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New post 18 Jul 2016, 08:17
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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Re: In the diagram, what is the length of AB? [#permalink]

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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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In the diagram, what is the length of AB? [#permalink]

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New post 18 Dec 2017, 06:06
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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Re: In the diagram, what is the length of AB? [#permalink]

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New post 18 Dec 2017, 06:24
geetikavig wrote:
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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Re: In the diagram, what is the length of AB? [#permalink]

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New post 31 Mar 2018, 23:16
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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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WhatsApp Image 2018-04-01 at 11.38.02.jpeg
WhatsApp Image 2018-04-01 at 11.38.02.jpeg [ 124.17 KiB | Viewed 79 times ]


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Re: In the diagram, what is the length of AB?   [#permalink] 31 Mar 2018, 23:16

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