Relationship between triangle and golden ratio

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Discussion Overview

The discussion revolves around the relationship between a 72-72-36 triangle and the golden ratio, exploring potential connections through geometric properties and trigonometric relationships.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant notes the triangle's angles relate to a pentagon, suggesting that the sides and diagonals of pentagons are connected to the golden ratio.
  • Another participant proposes examining the ratio of the long sides to the short side of the triangle using the sine rule to uncover the relationship with the golden ratio.
  • A later reply states that 2 sin(π/10) equals the golden ratio, implying a trigonometric connection.

Areas of Agreement / Disagreement

Participants express varying approaches to understanding the relationship, with no consensus reached on a definitive explanation or method.

Contextual Notes

Some assumptions about the geometric properties of the triangle and pentagon may not be fully explored, and the discussion relies on participants' recollections and interpretations of trigonometric relationships.

todd098
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I am having trouble finding the relationship between a 72-72-36 triangle and the golden ratio. Could someone point me in the right direction or explain it? Thanks
 
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Since 72 = 360/5, that triangle should be related to a pentagon, and I think the sides and the diagonals from pentagons are related by the golden ratio. I'm talking from memory, though, but it should be easy to check.
 
todd098 said:
I am having trouble finding the relationship between a 72-72-36 triangle and the golden ratio. Could someone point me in the right direction or explain it? Thanks

Did you ever consider looking at the ratio of the long sides to the short side of that triangle. Use the sine rule and I'm sure you'll find it easy enough.
 
Notice that 2 sin(Pi/10) = golden ratio
 

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