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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]

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21 Dec 2010, 08:17
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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
[Reveal] Spoiler: OA

Last edited by Bunuel on 23 Jan 2012, 03:05, edited 1 time in total.
Edited the question and added the image
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21 Dec 2010, 08:31
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In triangle ABC above, what is the length of side BC?

As <BDC=<BCD then the BD=BC. Also as <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=?

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

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26 Feb 2011, 22:07
the one point that puzzles me is how did you get /BAD = x ?
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27 Feb 2011, 00:52
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vibhav wrote:
the one point that puzzles me is how did you get /BAD = x ?

It's given that <BAD=x degrees (refer to the diagram in my first post on the page). Some OG books have a typo missing this info (as in vibhav post).
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27 Feb 2011, 17:56
thanks bunuel! this explains everything!
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23 Jan 2012, 00:36
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The question asks the length of side BC. From the figure, you can see that triangle BDC is an isosceles triangle with BD = BC. Thus, to know the length of BC, it is okay if we know the length of BD.

Statement 1:
To solve such problems, you have to know that in a triangle, the measure of the exterior angle is equal to the sum of the two non-adjacent angles of the triangle.

That is, in the given figure, for triangle ABD, angle BDC is the exterior angle. Thus, BDC = ABD + BAD
That is, 2x = ABD + x. Thus ABD = x.

Now, you can see that triangle ABD is an isosceles triangle in which AD = BD = 6. Thus, BD = BC = 6. SUFFICIENT

Statement 2:
x = 36 does not tell you anything about the length of any side. INSUFFICIENT

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Re: In triangle ABC above, what is the length of side BC? [#permalink]

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25 Jan 2012, 10:45
yeah its A. Bcoz B can only help us to find angle measure but no way we can fund side measrue from there .

so clearly A
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20 Mar 2012, 19:09
Using Similar triangles we know that
1) BD = BC (angle BDC = angle BCD)
2) We know angle BDA = 180-2x which means angle ABD = x
3) So, from the second similar triangle we know that angle BAD = angle ABD = x
4) Using similar triangles again; AD = BD
5) Combing 1 and 4; AD = BD = BC.

So, A is sufficient.

We don't really need the angle at all.
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Re: In triangle ABC above, what is the length of side BC? [#permalink]

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04 Apr 2012, 07:01
A

we know BD = BC

angle ADB = 180 - 2x

triangle ADB = (180-2x) + x + angle ABD

180 = 180 - 2x + x + angle ABD
x = angle ABD

therefore AD = BD = BC = 6

(B) x = 36
using this information we can only derive that AD = BD = BC , but we dont know value of side
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Re: In triangle ABC above, what is the length of side BC? [#permalink]

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22 Aug 2013, 05:13
Bumping for review and further discussion.
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Re: In triangle ABC above, what is the length of side BC? [#permalink]

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24 Jan 2014, 00:59
Bunuel wrote:
Bumping for review and further discussion.

Just a question. Would you have been able to solve for length BC if you were given another length? AB for example?
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Re: In triangle ABC above, what is the length of side BC? [#permalink]

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23 Apr 2014, 14:31
Why is AD=BD=BC? If AD is opposite to X, BD opposite to X as well and BC opposite to 2x?
Shouldn't BC be larger than both or equal to both TOGETHER?

Thanks for clarifying

Cheers!
J
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Re: In triangle ABC above, what is the length of side BC? [#permalink]

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23 Apr 2014, 22:48
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jlgdr wrote:
Why is AD=BD=BC? If AD is opposite to X, BD opposite to X as well and BC opposite to 2x?
Shouldn't BC be larger than both or equal to both TOGETHER?

Thanks for clarifying

Cheers!
J

Yes but they are sides of different triangles. Note that by the same logic, BD is opposite to 2x as well. The point is that it is opposite to x in one triangle (ABD) and opposite to 2x in another triangle (BDC).

BC will be equal to BD because they are both opposite 2x in triangle BDC.

AD will be equal to BD because they are both opposite angle x in triangle ABD.

So AD = BD = BC
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Re: In triangle ABC above, what is the length of side BC? [#permalink]

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20 Jun 2014, 03:35
Bunuel wrote:
Attachment:
trig2uc8.png
In triangle ABC above, what is the length of side BC?

As <BDC=<BCD then the BD=BC. Also as <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=?

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

I guess I need to expand my thought process...
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Re: In triangle ABC above, what is the length of side BC? [#permalink]

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08 Sep 2015, 19:20
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In triangle ABC above, what is the length of side BC? [#permalink]

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30 Sep 2015, 12:50
I'm sorry guys but the answer does not make sense at all!!

<BDC = <BCD ( I got this)
<ADB = 180 - 2x (I got this)
The sum of this triangle is 180

180 = x + (180-2x) + <ABD

now I'm confused. How did you come up with <ABD is equal to x??????? Even if we substitute <ABD with x, thus would be:

180 = x + 180-2x + x
180 = 2x -2x +180
180 = 180
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In triangle ABC above, what is the length of side BC? [#permalink]

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30 Sep 2015, 13:10
blendercroix wrote:
I'm sorry guys but the answer does not make sense at all!!

<BDC = <BCD ( I got this)
<ADB = 180 - 2x (I got this)
The sum of this triangle is 180

180 = x + (180-2x) + <ABD

now I'm confused. How did you come up with <ABD is equal to x??????? Even if we substitute <ABD with x, thus would be:

180 = x + 180-2x + x
180 = 2x -2x +180
180 = 180

The sum of two non-adjacent interior angles of a triangle is always equal to the measure of an exterior angle of a triangle. Even if you didn't know this property, say <ABD is y, so now we have <BAD is x, <ABD is y and <BDA is 180-2x. The sum of the interior angles of a triangle must sum to 180 degrees. So we have x+y+180-2x=180-->y=x
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Re: In triangle ABC above, what is the length of side BC? [#permalink]

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08 Oct 2015, 11:26
Just another take on the question. Experts can correct me.

< BAD is x and < BDC is 2x. It means triangle ABC can be considered as a triangle circumscribed in a circle with D as its center.
Hence, AD = DC = DB, the radius of the circle. Also, BD = BC (opposite to equal angles.)

Hence the length of AD is enough to answer the question. Choice A
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Re: In triangle ABC above, what is the length of side BC? [#permalink]

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11 Jan 2016, 13:50
anilnandyala wrote:
Attachment:
trig2uc8.png
In triangle ABC above, what is the length of side BC?

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

<BDC=2x and is the exterior angle for the triangle ABD, hence <ABD=x. AD=BD=BC

(2) x=36, clearly not sufficient.
(1) <BDC=2x and is the exterior angle for the triangle ABD, hence <ABD=x. AD=BD=BC=6. Sufficient

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Re: In triangle ABC above, what is the length of side BC? [#permalink]

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29 Jan 2016, 11:07
Bunuel wrote:
Attachment:
trig2uc8.png
In triangle ABC above, what is the length of side BC?

As <BDC=<BCD then the BD=BC. Also as <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=?

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

Why did you assume that ADC is a straight line. Because if it isint given specifically in the question then the entire logic fails. then angle BDA and BDC are not supplementary.
Re: In triangle ABC above, what is the length of side BC?   [#permalink] 29 Jan 2016, 11:07

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