Answer:
Perimeter of the given patio is [tex](\pi+6)y + 10x +8[/tex].
Step-by-step explanation:
Let us first label the given patio using the points A, B, C, D, E and F as shown the attachment diagram.
Perimeter of given figure can be found by the following:
Perimeter of polygon ABPCDEQF + Perimeter of semi-circle with diameter as EF.
Let us mark points P and Q on sides AB and FA respectively.
These points P and Q are just opposite to point D. Please refer to attached diagram for details.
As this figure has symmetry so, sides
[tex]side\ AP =side\ FE\ i.e\ 2y\\side\ AQ =side\ BC\ i.e.\ 2y\\side\ PB =side\ CD\ i.e.\ 3x+5\\side\ QF =side\ DE\ i.e.\ 2x-1[/tex]
Perimeter of polygon APBCDEQF = AP + PB + BC + CD + DE + FQ + QA (Side EF is not on the periphery so it is excluded)
[tex]\Rightarrow 2y + (3x+5) + 2y + (3x+5) + (2x - 1) + (2x-1) + 2y\\\Rightarrow 6y + 10x + 8 ...... (1)[/tex]
Perimeter of semicircle = [tex]\pi \times radius[/tex]
Radius of semicircle is y.
[tex]\text{Perimeter of semicircle = }\pi \times y ...... (2)[/tex]
Adding equations (1) and (2):
[tex]\text{Total perimeter = }6y + 10x + 8 + \pi y \\\Rightarrow (\pi+6)y + 10x + 8[/tex]
Perimeter of the given figure is [tex](\pi+6)y + 10x +8[/tex]
