Question 1201705
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As this problem is worded, printed, posted and presented, I do not understand, at all, what is about.


This combination of words "a triangle is joined with other triangle" is not a canonical term 
and has no a uniquely defined meaning in Math.


Edwin counted two ways to joint the given triangles by joining their legs.


There  is third way to join triangles by joining their hypotenuses.


In addition, there are several other ways to join triangles by joining a leg of one triangle and the hypotenuse of the other triangle.



Also, the words "joining triangles" does not necessary mean that the triangles' sides are congruent in any combination.



So, my impression that this "problem" is composed by a person who is unprofessional in writing Math problems,
but is more inclined to writing essays.


Writing Math and writing essays are two very different styles (like classic music and jazz are different styles), 
and I do not advise to mix these styles, if you want other people understand you and do not laugh quietly  to the side.



To the future generations of students, who, probably, will read this post, 
my advise is do not consider this problem seriously. Simply ignore it - as if you never saw it.



//////////////////



As an entertainment problem, it is, probably, not so bad.


If to consider it as entertainment problem, then the solution should be separated in parts.


<pre>
Part 1.  In 30-60-80 triangle the hypotenuse is 24 units long.
         Find the legs.


Part 2.  In 45-45-90 triangle one leg is 24 units.  
         Find the other leg and the hypotenuse.


Part 3.  In 45-45-90 triangle the hypotenuse is 24 units.  
         Find the legs.
</pre>

And several other similar parts.


The solution to each part is more than obvious.


Than more I submerge in this problem, than better I see that it is a perfect way 
to kill time for nothing, without any sense and without learning anything/nothing useful.



For young mind my advise is - keep away from this task; run out as fast as you can.



\\\\\\\\\\\\\\\\\\\



Edwin, if you read my post attentively, you will see that my pathos is directed
not against "the student" and not against your solution.


I highly appreciate your knowledge and your wisdom, your participation and your contribution in this forum.


My pathos is directed against those "writers", who try to compose Math problems 
and to disseminate them, without having adequate skills writing Math.



The number of such people in nova days is million (if not 100 millions).
They all do not know Math and can not express any single Math statement correctly. 
But they all want to be Math teachers and all want to teach others.