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## In the diagram, what is the length of AB?

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gmatter2012 Really wants to Beat The GMAT!
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In the diagram, what is the length of AB? Mon Jul 16, 2012 10:21 pm
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• Lap #[LAPCOUNT] ([LAPTIME])
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
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Last edited by gmatter2012 on Mon Jul 16, 2012 11:18 pm; edited 1 time in total

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niketdoshi123 Really wants to Beat The GMAT!
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Mon Jul 16, 2012 10:56 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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Mon Jul 16, 2012 11:01 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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Mon Jul 16, 2012 11:32 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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gmatter2012 Really wants to Beat The GMAT!
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Tue Jul 17, 2012 12:40 am
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

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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Tue Jul 17, 2012 8:10 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.

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.

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Tue Jul 17, 2012 9:02 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,

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,

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

Hope this helps!

gmatter2012 Really wants to Beat The GMAT!
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Tue Jul 17, 2012 10:08 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.

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.
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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Tue Jul 17, 2012 10:24 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.
Cheers!

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Wed Jul 18, 2012 12:25 am
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,

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.

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1-800-566-4043 (USA)

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