Answer:
part 32) [tex]P=2a+4b+3c\ units[/tex]
part 33) [tex]P=1.5x\ units[/tex]
Step-by-step explanation:
see the attached figure with letters to better understand the problem
Part 32) we know that
The perimeter of the figure is equal to the sum of its length sides
[tex]P=AB+BC+CD+DE+EF+FG+GH+HA[/tex]
we have
[tex]AB=2b+2c\\BC=c\\DE=a-c\\EF=b+c\\FG=3b\\GH=c\\HA=a-3b[/tex]
substitute the given values
[tex]P=(2b+2c)+c+CD+(a-c)+(b+c)+3b+c+(a-3b)[/tex]
Combine like terms
[tex]P=3b+4c+2a+CD[/tex]
Find the value of CD
we know that
AB=CD+EF+GH ----> by segment addition postulate
substitute
[tex]2b+2c=CD+b+c+c\\2b+c=CD+2c+b\\CD=b-c[/tex]
substitute in the formula of perimeter
[tex]P=3b+4c+2a+b-c\\P=2a+4b+3c[/tex]
Part 33) we know that
The perimeter of the figure is equal to the sum of its length sides
[tex]P=AB+BC+CD+DE+EF+FG+GH+HA[/tex]
we have
[tex]AB=\frac{x}{4}\\BC=\frac{x}{2}\\DE=\frac{x}{4}-y\\FG=y\\HA=\frac{x}{4}[/tex]
substitute the given values
[tex]P=\frac{x}{4}+\frac{x}{2}+CD+\frac{x}{4}-y+EF+y+GH+\frac{x}{4}[/tex]
Combine like terms
[tex]P=\frac{5}{4}x+CD+EF+GH[/tex]
we know that
[tex]AB=CD+EF+GH[/tex]----> by segment addition postulate
so
[tex]CD+EF+GH=AB=\frac{x}{4}[/tex]
substitute in the formula of perimeter
[tex]P=\frac{5}{4}x+\frac{x}{4}=\frac{6}{4}x[/tex]
Simplify
[tex]P=\frac{3}{2}x=1.5x\ units[/tex]
