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In the diagram, what is the length of AB?

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by niketdoshi123 » Mon Jul 16, 2012 9:56 pm
gmatter2012 wrote:In the diagram, what is the length of AB?


(1) BE = 3
(2) DE = 4
The sentence written above the triangle is not clear. If there is no relevant information in the sentence, then the solution goes as follows

Consider triangle BDE
BD = 5 ,which is hypotenuse (as angle(BED) = 90 )

Given in statement 1 BE = 3
=> DE = 4 (using Pythagoras' theorem)

statement 2 provides the same value of DE as we derived from statement 1.
Hence both the statement provide same information and thus we can eliminate options A, B ,and C.

As there is no other information given IMO the answer should be E
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by gmatter2012 » Mon Jul 16, 2012 10:01 pm
niketdoshi123 wrote:
gmatter2012 wrote:In the diagram, what is the length of AB?


(1) BE = 3
(2) DE = 4
The sentence written above the triangle is not clear. If there is no relevant information in the sentence, then the solution goes as follows

Consider triangle BDE
BD = 5 ,which is hypotenuse (as angle(BED) = 90 )

Given in statement 1 BE = 3
=> DE = 4 (using Pythagoras' theorem)

statement 2 provides the same value of DE as we derived from statement 1.
Hence both the statement provide same information and thus we can eliminate options A, B ,and C.

As there is no other information given IMO the answer should be E

The portion which is not clear states " In the diagram, what is the Length of AB ":)

The Question has now been edited to prevent further confusion, the irrelevant portion has been removed.
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by eagleeye » Mon Jul 16, 2012 10:32 pm
gmatter2012 wrote:In the diagram, what is the length of AB?


(1) BE = 3
(2) DE = 4


P.S. if using similarity please do show which two triangles are being taken and which are their corresponding sides
Refer to the diagram attached.

We are told that BD=5. We need to determine whether we can find AB (we don't actually have to find it).
Notice that, BED and ADB are similar triangles (We can infer this because of the three 90 degree angles in the RHS diagram that I drew.) since all three angles are the same. I have redrawn the similar triangles per their corresponding sides on LHS for clarity.

Hence ratio of corresponding sides are equal.
So, AB/BD = BD/DE = AD/BE. We know that BD = 5.
Hence AB/5 = 5/DE = AD/BE => AB= 25/DE.

Hence, if we know DE we can know what AB is.

With that in mind let's look at the statements:

1. BE = 3. In right angled triangle BED, if BD=5 and BE = 3, DE is definitely 4. (even without calculation, we can recognize the 3:4:5 right triangle). Since we know DE, we know AB. Sufficient.

2. DE = 4. Since we know DE, we can calculate AB. Sufficient again.

Hence D is correct.

:)
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by gmatter2012 » Mon Jul 16, 2012 11:40 pm
eagleeye wrote:
gmatter2012 wrote:In the diagram, what is the length of AB?


(1) BE = 3
(2) DE = 4


P.S. if using similarity please do show which two triangles are being taken and which are their corresponding sides
Refer to the diagram attached.

We are told that BD=5. We need to determine whether we can find AB (we don't actually have to find it).
Notice that, BED and ADB are similar triangles (We can infer this because of the three 90 degree angles in the RHS diagram that I drew.) since all three angles are the same. I have redrawn the similar triangles per their corresponding sides on LHS for clarity.

Hence ratio of corresponding sides are equal.
So, AB/BD = BD/DE = AD/BE. We know that BD = 5.
Hence AB/5 = 5/DE = AD/BE => AB= 25/DE.

Hence, if we know DE we can know what AB is.

With that in mind let's look at the statements:

1. BE = 3. In right angled triangle BED, if BD=5 and BE = 3, DE is definitely 4. (even without calculation, we can recognize the 3:4:5 right triangle). Since we know DE, we know AB. Sufficient.

2. DE = 4. Since we know DE, we can calculate AB. Sufficient again.

Hence D is correct.

:)

Nice solution

Are you ready for some discussion? ok here they come.( Please refer to your Attached diagram)

you have taken angle ABD = angle BDE How? This explanation has not been provided in the solution

which means you have taken AB parallel to DE.

Have you used the common sense that two perpendiculars drawn on the same straight line will be parallel to each other.Here BEC is the straight line and AB and DE are two perpendiculars drawn on it.

or have you used some other logic?
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by GMATGuruNY » Tue Jul 17, 2012 7:10 am
gmatter2012 wrote:In the diagram, what is the length of AB?


(1) BE = 3
(2) DE = 4


P.S. if using similarity please do show which two triangles are being taken and which are their corresponding sides
Each statement indicates that ∆BDE is a 3-4-5 triangle.
A height drawn through the right angle of a triangle forms SIMILAR TRIANGLES.
To keep track of the relationships, assign variables x and y to the unknown angles.

Image

Let ∠BAD = x and ∠ABD = y.
The sum of the interior angles of a triangle is 180.
x+y = 90.
Thus:
Any right triangle that includes x must also include y
Any right triangle that includes y must also include x.
The result is that all of the triangles in the figure above have the SAME COMBINATION OF ANGLES: x-y-90.
Thus, all of the triangles are similar.

With similar triangles, corresponding lengths are in the SAME RATIO.
Thus, when we compare ∆BDE to ∆ABD:

(side opposite x in ∆BDE/(side opposite 90 in ∆BDE) = (side opposite x in ∆ABD)/side opposite 90 in ∆ABD)
4/5 = 5/AB
AB = 25/4.

Thus, each statement is SUFFICIENT.
The correct answer is D.
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by imskpwr » Tue Jul 17, 2012 8:02 am
gmatter2012 wrote:In the diagram, what is the length of AB?


(1) BE = 3
(2) DE = 4


P.S. if using similarity please do show which two triangles are being taken and which are their corresponding sides
if BE = 3, so DE will always be 4
if DE = 4, so BE will always be 3
Hence, both statements are not required.........................one option gone.

Now,
AB = AD^2 + BD^2 = AD^2 + 25............so AD is required to answer

In BDC,
DE^2 = BE * EC................from here EC can be found out using anyone (1) or (2)

then,
In DEC,
DC can be found out.........as EC and DE are known now

In ABC,
BD^2 = AD * DC...............from here AD can be found out.

so any of the above option(1) or (2) is sufficient to answer.

Hope this helps!
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by gmatter2012 » Tue Jul 17, 2012 9:08 am
GMATGuruNY wrote:
gmatter2012 wrote:In the diagram, what is the length of AB?


(1) BE = 3
(2) DE = 4


P.S. if using similarity please do show which two triangles are being taken and which are their corresponding sides
Each statement indicates that ∆BDE is a 3-4-5 triangle.
A height drawn through the right angle of a triangle forms SIMILAR TRIANGLES.
To keep track of the relationships, assign variables x and y to the unknown angles.

Image

Let ∠BAD = x and ∠ABD = y.
The sum of the interior angles of a triangle is 180.
x+y = 90.
Thus:
Any right triangle that includes x must also include y
Any right triangle that includes y must also include x.
The result is that all of the triangles in the figure above have the SAME COMBINATION OF ANGLES: x-y-90.
Thus, all of the triangles are similar.

With similar triangles, corresponding lengths are in the SAME RATIO.
Thus, when we compare ∆BDE to ∆ABD:

(side opposite x in ∆BDE/(side opposite 90 in ∆BDE) = (side opposite x in ∆ABD)/side opposite 90 in ∆ABD)
4/5 = 5/AB
AB = 25/4.

Thus, each statement is SUFFICIENT.
The correct answer is D.
Thanks Mitch got it now basically in Triangle BDE angle BED 90 if angle BDE 90-x then angle DBE = x (1)

Now since angle ABE is also = 90
And angle ABE= angle ABD + angle DBE =90
and we know from 1 that angle DBE = x then angle ABD must be = 90-x
Hence we can see angle BDE=angle ABD =90-x

so in triangle ABD , angle ABD 90-x (from above) then angle BAD =x because addition of both should equal 90

hence now we know the angles which are equal and if we keep them in corresponding positions as eagle eye as shown then we get the corresponding sides then we can use the ratio and find AB

Thank you every one for your help.

p.s was just wondering isn't AB// DE in which case angle ABD = angle BDE alternate angles
The reason why they are parallel is because they make 90 degrees with the same straight line in the same direction.I think this is another way to find the angles which are equal, if there is a flaw here please let me know.
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by eagleeye » Tue Jul 17, 2012 9:24 am
gmatter2012 wrote: p.s was just wondering isn't AB// DE in which case angle ABD = angle BDE alternate angles
The reason why they are parallel is because they make 90 degrees with the same straight line in the same direction.I think this is another way to find the angles which are equal, if there is a flaw here please let me know.
Mitch gave a good explanation to how I did it, so I need not say anything else. As far as your query goes, AB is parallel to DE. Your reasoning for the parallelism is fine as well. And yes, you can use the parallel lines and supplementary/complementary angels approach to show that the aforementioned triangles are similar.
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by Anurag@Gurome » Tue Jul 17, 2012 11:25 pm
This problem can be simply solved by just looking at it if you know that
  • In a right-angled triangle, if a perpendicular is drawn from the right angle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle (and are also similar to each other.)
This can be proved as follows,
Image
In the triangle ABC, (angle BAD + angle BCD) = 90° ................ (1)
In the triangle ABD, (angle BAD + angle ABD) = 90° ................ (2)
In the triangle CBD, (angle CBD + angle BCD) = 90° ................ (3)

From (1) and (2) ---> angle BCD = angle ABD
From (1) and (3) ---> angle BAD = angle CBD

Hence, all the three triangles are similar triangles.


Now if we go on drawing perpendiculars to hypotenuse of the children triangles, all the triangles we will get will be similar to original triangle and all the other triangles.

Hence, in our question triangle BDE will be similar to triangle BAD. (This can be proved separately as others did but we don't need to do it as we know it has to be). Now, from the question stem we know the length of the side which is common to these two triangles. Hence, if we know the length of another side of triangle BDE, we can determine the length of AB.

Hence, both statements are individually sufficient.

The correct answer is D.
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by Anaira Mitch » Fri Dec 30, 2016 1:44 am
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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