1. Problem: Find the missing side length $x$ for each shape given the perimeter. (a) Rectangle with sides 8 cm and 5 cm, perimeter = 26 cm. Perimeter formula for rectangle: $$P=2(\text{length} + \text{width})$$ Given: $$26=2(8+5)$$ but one side is missing, so actually: $$26=8+5+5+x$$ Simplify: $$26=18+x$$ Solve for $x$: $$x=26-18=8$$ (b) Rectangle with sides 14 cm, 26 cm, and unknown $x$, perimeter = 80 cm. Perimeter formula: $$80=14+26+14+x$$ Simplify: $$80=54+x$$ Solve for $x$: $$x=80-54=26$$ (c) Square with three sides 5 cm and one side $x$, perimeter = 20 cm. Perimeter formula: $$20=5+5+5+x$$ Simplify: $$20=15+x$$ Solve for $x$: $$x=20-15=5$$ (d) Triangle with sides 9 cm, 9 cm, and $x$, perimeter = 25 cm. Perimeter formula: $$25=9+9+x$$ Simplify: $$25=18+x$$ Solve for $x$: $$x=25-18=7$$ (e) Equilateral triangle with all sides 12 cm, perimeter = 36 cm. Since all sides are equal: $$x=12$$ (f) Triangle with sides 22 cm, 22 cm, and $x$, perimeter = 79 cm. Perimeter formula: $$79=22+22+x$$ Simplify: $$79=44+x$$ Solve for $x$: $$x=79-44=35$$ (g) Parallelogram with sides 13.5 cm, 9 cm, and $x$, perimeter = 45 cm. Parallelogram opposite sides equal, so: $$x=9$$ Check perimeter: $$45=2(13.5+9)=2(22.5)=45$$ correct. (h) Regular pentagon with side $x$, perimeter = 2 m = 200 cm. Perimeter formula: $$200=5x$$ Solve for $x$: $$x=\frac{200}{5}=40$$ cm (i) Polygon with two sides 32 cm, 25 cm, and two sides $x$, perimeter = 163 cm. Perimeter formula: $$163=32+25+x+x$$ Simplify: $$163=57+2x$$ Solve for $x$: $$2x=163-57=106$$ $$x=\frac{106}{2}=53$$ Final answers: (a) $x=8$ cm (b) $x=26$ cm (c) $x=5$ cm (d) $x=7$ cm (e) $x=12$ cm (f) $x=35$ cm (g) $x=9$ cm (h) $x=40$ cm (i) $x=53$ cm