Given PQRS is a parallelogram, so PQ II SR and PS II QR
Also, AB II PS
To prove OC II SR
Proof In ΔOPS and ΔOAB,
PS II AB [given]
∠POS = ∠AOB [common angle]
∠OSP = ∠OBA [corresponding angles]
\(\therefore \Delta O P S \sim \Delta O A B\) [by AA similariry criterion]
Then, \(\frac{Q R}{A B}=\frac{C R}{C B}\)
\(\Rightarrow \frac{P S}{A B}=\frac{C R}{C B}\) ......(i)
\(\text { In } \Delta C Q R \text { and } \Delta C A B, Q R\|P S\| A B\)
∠QCR = ∠ACB [common angle]
∠CRQ = ∠CBA [corresponding angle]
∴ ΔCQR ∼ ΔCAB [by AA similariry criterion]
Then, \(\frac{Q R}{A B}=\frac{C R}{C B}\) 
\(\Rightarrow \frac{P S}{A B}=\frac{C R}{C B}\) ........(ii)
[since, PQRS is a parallelogram, so PS = QR]
From Eqs. (i) and (ii), we get
\(\frac{O S}{O B}=\frac{C R}{C B} \text { or } \frac{O B}{O S}=\frac{C B}{C R}\)
On subtracting 1 from both sides, we get
\(\frac{O B}{O S}-1=\frac{C B}{C R}-1\)
\(\Rightarrow \frac{O B-O S}{O S}=\frac{C B-C R}{C R} \Rightarrow \frac{B S}{O S}=\frac{B R}{C R}\)
By converse of basic proportionaliry theorem,
SR II OC

Prove \(\triangle DBA\) and \(\triangle DCB\) are similar.
\(\frac { DB }{ DA } =\frac { DC }{ DB } \)
\(\Rightarrow { DB }^{ 2 }=DA\times DC=4\times 5\)
\(\Rightarrow { DB }=2\sqrt { 5 } \) cm
In right \(\triangle BDC\) ,
BC² = BD² + CD² = \({ \left( 2\sqrt { 5 } \right) }^{ 2 }+{ \left( 5 \right) }^{ 2 }\)
BC = \(3\sqrt { 5 } \)
As, \(\triangle DBA\sim \triangle DCB\)
\(\Rightarrow \frac { DB }{ DC } =\frac { BA }{ BC } \)
\(\Rightarrow \frac { 2\sqrt { 5 } }{ 5 } =\frac { BA }{ 3\sqrt { 5 } } \)
BA = 6 cm
BA = 6 cm, BD = \(2\sqrt { 5 } \) cm

By BPT,
PQ : QR : RS = AB : BC : CD = 6 : 9 : 12
Let PQ = 6x, QR = 9x and RS = 12x
Since, length of PS = 36km
\(\therefore \)  PQ + QR + RS = 36
\(\Rightarrow \)  6x + 9x + 12x = 36
\(\Rightarrow \)  27x = 36  \(\Rightarrow \) x = \(\frac{4}{3}\)
PQ = 8 cm, QR = 12 cm and RS = 16 cm

Yes, 16 : 1 ; unity of nation, fraternity and patriotism.

Draw the figure according to given condition, then show that both triangles are similar by AAA similarity criterion and then use ratio of sides of both triangles to get the requried length.
10 m