Relationship between triangle and golden ratio

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SUMMARY

The relationship between a 72-72-36 triangle and the golden ratio is established through the properties of a pentagon. Specifically, the angles of the triangle correspond to the internal angles of a regular pentagon, where 72 degrees equals 360/5. The golden ratio can be derived by examining the ratio of the long sides to the short side of the triangle using the sine rule, where it is confirmed that 2 sin(π/10) equals the golden ratio.

PREREQUISITES
  • Understanding of basic trigonometry, specifically the sine rule.
  • Familiarity with the properties of regular pentagons.
  • Knowledge of the golden ratio and its mathematical significance.
  • Basic geometry concepts related to triangles and angles.
NEXT STEPS
  • Explore the properties of regular pentagons and their relationship to the golden ratio.
  • Study the sine rule in depth to understand its applications in triangle geometry.
  • Investigate the derivation of the golden ratio through trigonometric functions.
  • Learn about the geometric significance of 72-degree angles in various shapes.
USEFUL FOR

Mathematicians, geometry enthusiasts, and students studying trigonometry and the golden ratio will benefit from this discussion.

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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