Answer with explanation:
Polygon 1
AB=4.5 - 0.5=4
By distance formula
[tex]BC=\sqrt{(2.5-0.5)^2+(6.5-4.5)^2}\\\\BC=\sqrt{4+4}\\\\BC=2 \sqrt{2}\\\\CD=\sqrt{(2.5-3.5)^2+(6.5-4.5)^2}\\\\CD=\sqrt{1+4}\\\\CD=\sqrt{5}\\\\DE=4,\\\\AE=3.5-0.5=3[/tex]
Polygon 2
G H=F J=G F=2 units,
[tex]HI=\sqrt{(6-7)^2+(2.5-1)^2}\\\\HI=\sqrt{1+2.25}\\\\NO=\sqrt{3.25}\\\\JI=\sqrt{(7-6)^2+(1-0.5)^2}\\\\JI=\sqrt{1.25}[/tex]
Polygon 3
LM=3 units,
[tex]MN=\sqrt{(8-7)^2+(4.5-5)^2}\\\\MN=\sqrt{1+(-0.5)^2}\\\\MN=\sqrt{1.25}\\\\NO=\sqrt{(8-7)^2+(4.5-3)^2}\\\\NO=\sqrt{1+2.25}\\\\NO=\sqrt{3.25}\\\\OK=3\\\\KL=2[/tex]
Polygon 4
PQ=9-7.5=1.5
PT=QR=3.5-1.5=2
[tex]TS=\sqrt{(8-7.5)^2+(1.5-0.5)^2}\\\\TS=\sqrt{0.25+1}=\sqrt{1.25}\\\\SR=\sqrt{(8-9)^2+(1.5-0.5)^2}\\\\RS=\sqrt{1+1}=\sqrt{2}[/tex]
Now we will check which polygon have corresponding proportional sides.
→→→Corresponding sides of polygon 1 and Polygon 4
[tex]\frac{DE}{TP}=\frac{AE}{PQ}=\frac{AB}{QR}=\frac{BC}{RS}=\frac{CD}{ST}\\\\=\frac{4}{2}=\frac{3}{1.5}=\frac{4}{2}=\frac{2\sqrt{2}}{\sqrt{2}}\neq \frac{\sqrt{5}}{1.25}[/tex]
Not Similar.
→→Corresponding sides of polygon 1 and Polygon 2
[tex]\frac{DE}{FJ}\neq \frac{AE}{GF}\\\\ \frac{4}{2}\neq\frac{3}{2}[/tex]
Not Similar.
→→Corresponding sides of polygon 1 and Polygon 3
[tex]\frac{DE}{KO}\neq \frac{AE}{KL}\\\\\frac{4}{3}\neq\frac{3}{2}[/tex]
Not Similar.
→→Corresponding sides of polygon 2 and Polygon 4
[tex]\frac{FJ}{TP}\neq\frac{GF}{PQ}\\\\\frac{2}{2}\neq\frac{2}{1.5}[/tex]
Not Similar.
→→⇒None of the four given Option are true.
