The discussion revolves around the possibility of forming a triangle with sides defined as \(3\), \(3r\), and \(3r^2\), where \(r\) is a real number greater than the Golden Ratio. Participants explore the implications of the triangle inequality theorem in this context.
The discussion includes various interpretations of the problem, with some participants questioning the validity of the assumptions made about \(r\). There are attempts to derive inequalities to determine the feasibility of forming a triangle, and some participants indicate a shift in understanding as they engage with the mathematical reasoning.
There is mention of the Golden Ratio and its significance in the context of the problem. The triangle inequality theorem is referenced as a critical component in evaluating the possibility of forming a triangle with the given side lengths.
Nipuna Weerasekara said:Homework Statement
There is a triangle with sides $$ 3,3r,3r^2 $$ such that 'r' is a real number strictly greater than the Golden Ratio.
Is this statement true or false...?
Homework Equations
$$Golden \space Ratio = \phi = 1.618... $$
The Attempt at a Solution
Actually I have no clue at all how to approach to this kind of question.
I guess soMath_QED said:When I give you 3 line segments with a random length, can you make a triangle with it?
Is,'t that trvially satisfied?Nipuna Weerasekara said:Eventually one inequality leads that r has a complex variance. (by solving $$3r^2 +3 > 3r$$)
I don't get what you say here.micromass said:Is,'t that trvially satisfied?
So ##r=10## is obviously greater than ##\phi## and you've found a triangle with sides ##3, 30## and ##300##?Nipuna Weerasekara said:The statement is true. All r is real and greater than Golden Ratio (1.61803)...