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

Question

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.

viktorija · Answer

(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.
Answer C

scorpionkapoor77 · Answer

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

Mehemmed · Answer

Bunuel , could you please provide a detailed solution for this question?

fskilnik · Answer

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.

fskilnik · Answer

\angle ABC = {105^ \circ }

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

https://preview.ibb.co/jf031z/01_Oct18_3w.gif

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

https://preview.ibb.co/k0BCTe/01_Oct18_4q.gif

(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.

ShankSouljaBoi · Answer

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

chetan2u · Answer

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

ShankSouljaBoi · Answer

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 ?

Posted from my mobile device

bumpbot · Answer

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

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