Geometry & Algebra: Proving Triangle-Square

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In summary, the conversation discusses the concept of a square fitting into a triangle and how it can impact the lengths of the triangle's sides. Some participants argue that if a square fits perfectly into a triangle, at least two sides of the triangle must be of equal length. This is explained geometrically and algebraically, with a counterexample provided. Eventually, it is suggested that the square must line up with the midpoint of one of the sides for this statement to hold true. The conversation concludes with a final clarification and agreement on this condition.
  • #1
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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  • #2
Is this a homework question? And if not, an you show us or explain to us what you tried?
 
  • #3
I'm not convinced that this is true. Why doesn't my counterexample work?
 

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  • counterexample.png
    counterexample.png
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  • #4
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?
 
  • #5
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.
 
  • #6
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.
 
  • #7
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.
 
  • #8
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
 
  • #9
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!
 

Related to Geometry & Algebra: Proving Triangle-Square

What is the difference between a triangle and a square?

A triangle is a polygon with three sides and three angles, while a square is a polygon with four equal sides and four right angles.

How can I prove that a triangle and square are congruent?

To prove that a triangle and square are congruent, you can use any of the following methods:

  • Side-Side-Side (SSS) Congruence: If all three sides of a triangle are equal to the corresponding sides of a square, then the two shapes are congruent.
  • Angle-Angle-Side (AAS) Congruence: If two angles and the included side of a triangle are equal to two angles and the included side of a square, then the two shapes are congruent.
  • Side-Angle-Side (SAS) Congruence: If two sides and the included angle of a triangle are equal to two sides and the included angle of a square, then the two shapes are congruent.

What is the Pythagorean theorem and how is it used in proving triangle-square congruence?

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is often used in proving triangle-square congruence by showing that the three sides of a triangle are equal to the corresponding sides of a square, and therefore the two shapes are congruent.

Can a triangle and a square be similar but not congruent?

Yes, a triangle and a square can be similar but not congruent. Similar shapes have the same shape, but not necessarily the same size. This means that the corresponding angles of the two shapes are equal, but the corresponding sides are proportional.

What practical applications does knowing how to prove triangle-square congruence have?

Knowing how to prove triangle-square congruence can be useful in a variety of fields, including engineering, architecture, and computer graphics. For example, in engineering and architecture, congruent shapes are important for creating structures that are stable and structurally sound. In computer graphics, the ability to prove congruence between shapes is essential for creating accurate and realistic 3D models.

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