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# In triangle ABC above, what is the length of side BC?

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In triangle ABC above, what is the length of side BC? [#permalink]  03 Mar 2014, 23:20
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Difficulty:

65% (hard)

Question Stats:

59% (02:25) correct 41% (01:59) wrong based on 445 sessions
The Official Guide For GMAT® Quantitative Review, 2ND Edition

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In triangle ABC above, what is the length of side BC?

(1) Line segment AD has length 6.
(2) x = 36

Data Sufficiency
Question: 123
Category: Geometry Triangles
Page: 162
Difficulty: 650

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Math Expert
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Re: In triangle ABC above, what is the length of side BC? [#permalink]  03 Mar 2014, 23:21
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SOLUTION

In triangle ABC above, what is the length of side BC?

Since <BDC = <BCD then the BD=BC. Also, since <ADB = 180 -2x (exterior angle) and the sum of the angles of a triangle is 180 degrees, then in triangle ADB we'll have: x + (180 - 2x) + <ABD = 180 --> <ABD = x. Now, we have that <ABD = x = <DAB so AD = BD --> AD = BD = BC.

Question: BC=?

(1) Line segment AD has length 6 --> AD = BD = BC = 6. Sufficient.

(2) x = 36 --> we know only angles which is insufficient to get the length of any line segment.

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Re: In triangle ABC above, what is the length of side BC? [#permalink]  05 Mar 2014, 21:39
Is there a way to tell if D is the bisector of AC?

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Re: In triangle ABC above, what is the length of side BC? [#permalink]  05 Mar 2014, 23:03
1
KUDOS
angle ABD =x, so AD=DB=BC, A is Sufficient
B gives only angle values and only angle values can not give length of any side

only A is suff
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Re: In triangle ABC above, what is the length of side BC? [#permalink]  05 Mar 2014, 23:49
1
KUDOS
Given data:-
In Triangle ABC:-
Angle A+ Angle ABC + Angle C = 180
Thus Angle B = 180-3x

In Triangle DBC:-
Angle D + Angle DBC + Angle C = 180
Thus Angle DBC = 180 - 4x

Thus Angle ABD = Angle ABC - Angle DBC = 180 - 3x - (180 - 4x) = x

In Triangle ABD
Angle ABD = Angle A = x
Thus the opposite sides are equal. i.e AD = DB ....1

From Triangle DBC
Since Angle D = Angle C
BD = BC .....2

Question value of BC?
Thus if we find the value of either AD or BD we can get value of BC.

There is our answer. Since BC=AD this BC = 6. Statement is sufficient.

St2 :- x=36
From this angle value we cant get the value of any line segment. hence it is insufficient.

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Re: In triangle ABC above, what is the length of side BC? [#permalink]  08 Mar 2014, 13:04
Expert's post
SOLUTION

In triangle ABC above, what is the length of side BC?

Since <BDC = <BCD then the BD=BC. Also, since <ADB = 180 -2x (exterior angle) and the sum of the angles of a triangle is 180 degrees, then in triangle ADB we'll have: x + (180 - 2x) + <ABD = 180 --> <ABD = x. Now, we have that <ABD = x = <DAB so AD = BD --> AD = BD = BC.

Question: BC=?

(1) Line segment AD has length 6 --> AD = BD = BC = 6. Sufficient.

(2) x = 36 --> we know only angles which is insufficient to get the length of any line segment.

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Re: In triangle ABC above, what is the length of side BC? [#permalink]  08 Mar 2014, 16:48
This was a super easy question...I said D because I wasn't paying attention....PEOPLE....PAY ATTENTION!!!!
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Re: In triangle ABC above, what is the length of side BC? [#permalink]  23 Apr 2014, 05:53
I solved this question in the following way:

At first sight you must see that <DBC and <CDB have the same angle.
So the length of BD must be equal to BC.
The rephrased question is then as followings: What is BD?

First analyse the given picture as 1 triangle (ABC)
The angle of B would be : 180 - Angle A - Angle C = 180 - 3x

Secondly look at the triangle (BCD).
The angle of B would be now: 180 - Angle C - Angle D = 180 - 4x

To find the the Angle of B in the Triangle of ABD , subtract these two equations:
180 - 3x - 180 --4x(=+)= x

Since Angle <ABD = <BDA
The length of AD must be equal to BD. ( Because they have the same angle )

If you know the length of AD or BD , you have sufficient information to answer the question.

(1) Sufficient , you now know the length of AD.
(2) You know nothing about any length , clearly insufficient.
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In triangle ABC above, what is the length of side BC? [#permalink]  17 May 2015, 06:56
Bunuel wrote:
SOLUTION

In triangle ABC above, what is the length of side BC?

Since <BDC = <BCD then the BD=BC. Also, since <ADB = 180 -2x (exterior angle) and the sum of the angles of a triangle is 180 degrees, then in triangle ADB we'll have: x + (180 - 2x) + <ABD = 180 --> <ABD = x. Now, we have that <ABD = x = <DAB so AD = BD --> AD = BD = BC.

Question: BC=?

(1) Line segment AD has length 6 --> AD = BD = BC = 6. Sufficient.

(2) x = 36 --> we know only angles which is insufficient to get the length of any line segment.

Hey, great explaination, thanks.

Question: Is it the case that whenever angles are equal, their sides must equal?

Is there no exception where angle a $$=$$ angle b but length a $$=/=$$ length b?
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Re: In triangle ABC above, what is the length of side BC? [#permalink]  18 May 2015, 03:33
Expert's post
erikvm wrote:
Bunuel wrote:
SOLUTION

In triangle ABC above, what is the length of side BC?

Since <BDC = <BCD then the BD=BC. Also, since <ADB = 180 -2x (exterior angle) and the sum of the angles of a triangle is 180 degrees, then in triangle ADB we'll have: x + (180 - 2x) + <ABD = 180 --> <ABD = x. Now, we have that <ABD = x = <DAB so AD = BD --> AD = BD = BC.

Question: BC=?

(1) Line segment AD has length 6 --> AD = BD = BC = 6. Sufficient.

(2) x = 36 --> we know only angles which is insufficient to get the length of any line segment.

Hey, great explaination, thanks.

Question: Is it the case that whenever angles are equal, their sides must equal?

Is there no exception where angle a $$=$$ angle b but length a $$=/=$$ length b?

Yes, the base angles of an isosceles triangle are always equal and vise-versa: if two angles in a triangle are equal then it's an isosceles triangle.
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Re: In triangle ABC above, what is the length of side BC? [#permalink]  12 Jul 2015, 03:07
Hi All, I solved it the same way as Bunuel stated... so after some rearangements for St1 it's sufficient to know that ad=6, and St2 is not sufficient, because we have only degrees and we need length...

But after solving this question I have still one question: we say that AD=BD, but how can it be that a side opposite to a smaller angle X° is equal to the side opposite to a larger angle 2X° (BCD), istn't it a bit weird ??
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Re: In triangle ABC above, what is the length of side BC? [#permalink]  12 Jul 2015, 03:36
BrainLab wrote:
Hi All, I solved it the same way as Bunuel stated... so after some rearangements for St1 it's sufficient to know that ad=6, and St2 is not sufficient, because we have only degrees and we need length...

But after solving this question I have still one question: we say that AD=BD, but how can it be that a side opposite to a smaller angle X° is equal to the side opposite to a larger angle 2X° (BCD), istn't it a bit weird ??

IMO, your statement of side opposite x deg will be smaller than the side opposite 2x degrees is ONLY applicable for the same triangle. I can have 2 different triangles with x and 2x degrees and the corresponding 'opposite' sides still being the same (I can modify the other angles or the proportion of the other 2 sides to counter the effect of the additional 'x' degrees!)

But an interesting question.
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Re: In triangle ABC above, what is the length of side BC? [#permalink]  05 Sep 2015, 16:09
Only 55% got right ( at time of writing) and average time for solving it right is 2:20. A similar performing question else where is generally 95% hard and is 700 range question.

How is the difficulty decided?
Re: In triangle ABC above, what is the length of side BC?   [#permalink] 05 Sep 2015, 16:09
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