We have the parallel sides of the rectangle are equal, therefore:
[tex]\begin{gathered} RS=QP=4x+3 \\ \text{and} \\ SP=RQ=5x \end{gathered}[/tex]
The perimeter is the sum of all sides, then:
[tex]RS+QP+SP+RQ=222[/tex]
Substitute the given data:
[tex](4x+3)+(4x+3)+5x+5x=222[/tex]
And solve for x:
[tex]\begin{gathered} 4x+3+4x+3+5x+5x=222 \\ 18x+6=222 \\ 18x+6-6=222-6 \\ 18x=216 \\ \frac{18x}{18}=\frac{216}{18} \\ x=12 \end{gathered}[/tex]
Next, we find the length of side RS:
[tex]RS=4x+3=4(12)+3=48+3=51[/tex]
Answer: RS = 51 units
