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In triangle ABC, angle ABC is 105 degrees. Is the area of triangle ABC

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In triangle ABC, angle ABC is 105 degrees. Is the area of triangle ABC  [#permalink]

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05 Feb 2015, 08:58
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In triangle ABC, angle ABC is 105 degrees. Is the area of triangle ABC less than 5?

(1) Segment AB = 2√2
(2) Angle BCA is 30 degrees.

Kudos for a correct solution.

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Re: In triangle ABC, angle ABC is 105 degrees. Is the area of triangle ABC  [#permalink]

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05 Feb 2015, 19:54
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(1) Not sufficient.
(2) Not sufficient. Just knowing the values of all angles does not allow us to find the area.

(1)+(2) Sufficient. From (1) we know that AB is 2√2, so if we spit the triangle ABC into two triangles drawing a line from the angle ABC, we can get a two special right triangles. First triangle 45-45-90 with a hypotenuse 2√2 and the second triangle 30-60-90. From that point we can find all the needed values.
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Re: In triangle ABC, angle ABC is 105 degrees. Is the area of triangle ABC  [#permalink]

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25 Aug 2015, 13:22
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I don't think we require splitting and spend time on it. Once we know all three angles and length of one side, other sides can be known through the ratio of these angles. However it is not essential but the formula to find the lengths is a/sin A = b/Sin B = c/Sin C
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Re: In triangle ABC, angle ABC is 105 degrees. Is the area of triangle ABC  [#permalink]

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05 Jan 2018, 06:43
Bunuel, could you please provide a detailed solution for this question?
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Re: In triangle ABC, angle ABC is 105 degrees. Is the area of triangle ABC  [#permalink]

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01 Oct 2018, 12:36
Mehemmed wrote:
Bunuel, could you please provide a detailed solution for this question?

Hi, Mehemmed!

I will take the liberty to post my detailed solution below.
Please feel free to ask me (or Bunuel) about it!

Regards and success in your studies,
Fabio.
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Re: In triangle ABC, angle ABC is 105 degrees. Is the area of triangle ABC  [#permalink]

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01 Oct 2018, 12:40
Bunuel wrote:
In triangle ABC, angle ABC is 105 degrees. Is the area of triangle ABC less than 5?

(1) Segment AB = 2√2
(2) Angle BCA is 30 degrees.

$$\angle ABC = {105^ \circ }$$

$${S_{\Delta ABC}}\,\,\mathop < \limits^? \,\,\,5$$

(2) We have two (therefore all three) internal angles of triangle ABC.
This is not enough:

(1+2) We "know" the triangle ABC! I mean, all its angles, all its side lengths.

Reason: any two triangles that satisfy the question stem (pre-statements) and statement (2) are similar.
When we add statement (1) info, we are able to find the "scale" involved, hence any two triangles that satisfy
the question stem AND both statements together are CONGRUENT. In a sense, there is just one triangle ABC!

Conclusion: the area of triangle ABC is UNIQUE, hence our answer is also UNIQUE. Sufficient!

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: In triangle ABC, angle ABC is 105 degrees. Is the area of triangle ABC  [#permalink]

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02 Oct 2018, 10:45
Property: Two angles and one side can give you a unique triangle. By this I mean you can find each angle and each length.
1. Length of the triangle known and just one angle. Insuff.
2. One more angle given. Insuff.
Combining
One Length and Two angles known . Therefore you have a unique or a fixed triangle and consequently you can find all the dimensions of the triangle. Hence, be Herons formula you can eventually calculate the area, which will be either greater than 25 or lesser than 25(need not compute).
Sufficient

chetan2u Can you please guide me with this approach ? If this sounds fine to you.

Thanks
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In triangle ABC, angle ABC is 105 degrees. Is the area of triangle ABC  [#permalink]

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02 Oct 2018, 11:47
ShankSouljaBoi wrote:
Property: Two angles and one side can give you a unique triangle. By this I mean you can find each angle and each length.
1. Length of the triangle known and just one angle. Insuff.
2. One more angle given. Insuff.
Combining
One Length and Two angles known . Therefore you have a unique or a fixed triangle and consequently you can find all the dimensions of the triangle. Hence, be Herons formula you can eventually calculate the area, which will be either greater than 25 or lesser than 25(need not compute).
Sufficient

chetan2u Can you please guide me with this approach ? If this sounds fine to you.

Thanks

Yes. If you can get a unique triangle, you can find its area .....
It may be whatever way, but literally measuring it or through angles , doesn't matter till you have a unique figure that has a constant area..

Now COMBINED..
I have two angles, of course third angle can ne known as the sum of all is 180..
So we know all angles and one side, you cannot make more than one triangle of this type...
So you have a unique triangle and unique area..
Sufficient

C
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Re: In triangle ABC, angle ABC is 105 degrees. Is the area of triangle ABC  [#permalink]

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02 Oct 2018, 11:52
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chetan2u In the last paragraph you have mentioned *you can make more than one triangle* I did not understand this. We're you writing can not ?

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Re: In triangle ABC, angle ABC is 105 degrees. Is the area of triangle ABC   [#permalink] 02 Oct 2018, 11:52
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