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

**, is called**

*Regular Pentagon***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