Geometry & Algebra: Proving Triangle-Square

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SUMMARY

The discussion centers on the geometric and algebraic proof that a square can fit within a triangle, specifically requiring that at least two sides of the triangle must be of equal length. Participants debate the conditions under which this is true, suggesting that the square must align with the midpoint of one of the triangle's sides for the assertion to hold. The conversation highlights the necessity of visual representation to clarify configurations and emphasizes the importance of symmetry in geometric proofs.

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  • Familiarity with triangle classifications (isosceles, scalene)
  • Knowledge of geometric proofs and symmetry
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adjacent
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As you see in the attachment if a square exactly fits in a triangle of any shape,Then atleast Two sides of that triangle would be of equal length(Upper two sides as in attachment)

Explain this Geometrically and algebraically

I tried it but in vain(Im a 9th grader)
 

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  • Square-Triangle.png
    Square-Triangle.png
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Is this a homework question? And if not, an you show us or explain to us what you tried?
 
I'm not convinced that this is true. Why doesn't my counterexample work?
 

Attachments

  • counterexample.png
    counterexample.png
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adjacent said:
As you see in the attachment if a square exactly fits in a triangle of any shape,Then atleast Two sides of that triangle would be of equal length(Upper two sides as in attachment)

Explain this Geometrically and algebraically

I tried it but in vain(Im a 9th grader)

Well, see, here's the thing. When you CHOOSE a triangle that has two equal sides, you do indeed get a triangle with 2 equal sides. So what? What about Number Nine's example?
 
My bet is that part of the assignment details that the square has to line up with the midpoint of one of the sides. Then (I think) it would be true.
 
Vorde said:
My bet is that part of the assignment details that the square has to line up with the midpoint of one of the sides. Then (I think) it would be true.

It would almost certainly be true in that case. I would accept a handwavy proof by symmetry.
 
sorry!Only one side of the square should touch a side of the triangle.The other two sides should touch one of the corners of the square.Then this will be correct.This is no homework,I am just curious.I could not find any explanation.
 
adjacent said:
sorry!Only one side of the square should touch a side of the triangle.The other two sides should touch one of the corners of the square.Then this will be correct.This is no homework,I am just curious.I could not find any explanation.

What do you mean "touch a side"? It's not clear what configuration you mean. Draw an example
 
adjacent said:
sorry!Only one side of the square should touch a side of the triangle.The other two sides should touch one of the corners of the square.Then this will be correct.This is no homework,I am just curious.I could not find any explanation.

I'm still not convinced. You could still have a situation like the one illustrated in my picture (attached), and I'm not sure that 2 sides would necessarily be the same length.
 

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  • counterexample.png
    counterexample.png
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  • #10
You are right Number nine,Thanks!
 

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