Golden Ratio and The Pentagon (5 Powerful Geometry Hack )

Two quantities a & b are in the “Golden Ratio”, if (a+b): a =a: b, where a>b>0. It is represented by ‘𝞅‘.

𝞅=(1+√5)/2=1.618033988749….

It is also known as the Divine Proportion and the extreme and mean ratio.

Pentagram: A 5 Point Star, formed from the diagonals of a Regular Pentagon, is called Pentagram or Pentangle.

In the given figure, we have to prove that PS : PR=PR: PQ= PQ: QR= 𝞅=1.618….

Internal Angle of the Regular Polygon =(n-2)π/n

Where ‘n’ is the number of sides and π=180°

Internal Angle=(5-2)×180°/5=108°

∠PAS=108°

∠APS=∠ASP=36°   

In ∆PAS, apply Cosine Formula 

PS²=AP²+AS² – 2*AP*AS*Cos(∠PAS) 

      =x²+x² -2x² Cos(108°) 

      =2x²{ 1 – (1-√5)/4}

      =x²(3+√5)/2

      =(6+2√5)x²/4

      =(√5+1)²x²/4

PS=(√5+1)/2  ………..(i)

Similarly, Apply the Cosine formula in ∆ARS

k=(√5-1)x/2

PQ=RS=k=(√5-1)x/2

So, QR= PS-2k =(√5+1)/2 -2*(√5-1)/2 =(3-√5)/2

QR=(3-√5)/2

PQ: QR: RS=(√5-1) : (3-√5): (√5-1)

So, PS=(√5-1)+(3-√5)+(√5-1)=√5+1

PR=(√5-1)+(3-√5)=2

PS: PR=(1+√5):2  

PR: PQ= 2: (√5-1)=(1+√5):2  

PQ: QR=(√5-1) : (3-√5)=(1+√5):2   ……………..{Multiply (3+√5) in numerator and denominator}

Hence Proved…

The Centroid of the Triangle (All Concepts): Click Here