Two quantities a & b are in the βGolden Ratioβ, if (a+b): a =a: b, where a>b>0. It is represented by βπ β.
π =(1+β5)/2=1.618033988749β¦.
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= π =1.618β¦.
Internal Angle of the Regular Polygon =(n-2)Ο/n
Where βnβ is the number of sides and Ο=180Β°
Internal Angle=(5-2)Γ180Β°/5=108Β°
β PAS=108Β°
β APS=β ASP=36Β°
In βPAS, apply Cosine Formula
PSΒ²=APΒ²+ASΒ² β 2*AP*AS*Cos(β PAS)
=xΒ²+xΒ² -2xΒ² Cos(108Β°)
=2xΒ²{ 1 β (1-β5)/4}
=xΒ²(3+β5)/2
=(6+2β5)xΒ²/4
=(β5+1)Β²xΒ²/4
PS=(β5+1)/2 β¦β¦β¦..(i)
Similarly, Apply the Cosine formula in βARS
k=(β5-1)x/2
PQ=RS=k=(β5-1)x/2
So, QR= PS-2k =(β5+1)/2 -2*(β5-1)/2 =(3-β5)/2
QR=(3-β5)/2
PQ: QR: RS=(β5-1) : (3-β5): (β5-1)
So, PS=(β5-1)+(3-β5)+(β5-1)=β5+1
PR=(β5-1)+(3-β5)=2
PS: PR=(1+β5):2
PR: PQ= 2: (β5-1)=(1+β5):2
PQ: QR=(β5-1) : (3-β5)=(1+β5):2 β¦β¦β¦β¦β¦..{Multiply (3+β5) in numerator and denominator}
Hence Provedβ¦
The Centroid of the Triangle (All Concepts): Click Here