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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.

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.

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

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23 Apr 2014, 06:53

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 )

AD=BD=BC

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]

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17 May 2015, 07:56

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.

Answer: A.

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?

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.

Answer: A.

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]

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12 Jul 2015, 04:07

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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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!)

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

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05 Sep 2015, 17:09

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.

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

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24 Sep 2016, 08:21

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

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18 Nov 2016, 12:13

Engr2012 wrote:

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.

Had the same question as brainlab did. That was throwing me off.
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Re: In triangle ABC above, what is the length of side BC? [#permalink]

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23 Feb 2017, 05:44

PROMPT ANALYSIS

The figure has angle BAD =x, angle BDC =2x, angle BCD = 2x.

Super set The side length of BC could be any positive real number.

Translation Since Angle BAD + Angle ABD = Angle BDC therefore angle ABD = x. Hence triangle ABD and triangle BDC are isosceles triangles. Hence AD = BD = BC. In order to find the length of BC we need: 1# exact value of BC. 2# relation or property that will lead us to find the length of BC.

Statement analysis St 1: AD = 6. Since AD = BD = BC, therefore BC = 6 ANSWER. Option b, c and e. St 2: x =36. Since there is no idea about any side of the the figure, therefore it is insufficient.

Hence option A

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Re: In triangle ABC above, what is the length of side BC?
[#permalink]
23 Feb 2017, 05:44

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