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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=?

(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.

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

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23 Jan 2012, 00:36

2

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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

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

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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.

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.

Re: 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

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

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08 Oct 2015, 11:26

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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
_________________

KudosPlease if you find my question / solution helpful.

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=?

(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.

Answer: A.

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.

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=?

(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.

Answer: A.

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.

OFFICIAL GUIDE:

Problem Solving Figures: All figures accompanying problem solving questions are intended to provide information useful in solving the problems. Figures are drawn as accurately as possible. Exceptions will be clearly noted. Lines shown as straight are straight, and lines that appear jagged are also straight. The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero. All figures lie in a plane unless otherwise indicated.

Data Sufficiency: Figures: • Figures conform to the information given in the question, but will not necessarily conform to the additional information given in statements (1) and (2). • Lines shown as straight are straight, and lines that appear jagged are also straight. • The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero. • All figures lie in a plane unless otherwise indicated.
_________________

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=?

(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.

Answer: A.

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.

The lines that appear straight are straight. Also ABC is a triangle (given). So AC is a straight line. D is a point on AC and we have been given the measure of angle BDC as 2x. Hence there is no ambiguity here.
_________________

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

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04 Mar 2016, 11:01

Took me some time to see that this question is simple:

Concept: In a triangle, the the sum of two angles = the measure of the outside angle from the last angle. To be clear: 2x = x + (the other angle), which can only be x.

(1) We have that AD = 6. Using the concept, we can see that the triangle is isosceles and share one side with another isosceles triangle. So AD=BD=BC SUFFICIENT.

(2) INSUFFICIENT: we have no values to find any measure.