Question

find the perimeter of each polygon.

1. rectangle with sides 4 in. and 8 in.
2. regular octagon with side 9 cm
3. polygon with side - lengths 7 mm, 8 mm, 4 mm, 4 mm, 5 mm

find the value of x in each polygon.

1. rectangle with width 3 cm, length 2x, and perimeter p = 22 cm
2. regular octagon with perimeter p = 80 ft and side x
3. composite - shaped polygon with given side - lengths and perimeter p = 30 m

Explanation:

Step1: Recall perimeter formula for rectangle

For a rectangle with length $l$ and width $w$, the perimeter $P = 2(l + w)$. In problem 1, $l = 8$ in and $w = 4$ in. So $P=2(8 + 4)$.

Step2: Calculate perimeter of rectangle in problem 1

$P=2\times12=24$ in.

Step3: Recall perimeter formula for regular octagon

For a regular octagon with side - length $s$, the perimeter $P = 8s$. In problem 2, $s = 9$ cm. So $P = 8\times9$.

Step4: Calculate perimeter of octagon in problem 2

$P=72$ cm.

Step5: Calculate perimeter of the polygon in problem 3

Add up the side - lengths: $P=7 + 6+4 + 4+5=26$ mm.

Step6: Use perimeter formula for rectangle to find $x$ in problem 4

For the rectangle with length $2x$ and width $3$ cm and $P = 22$ cm, we use $P = 2(l + w)$. So $22=2(2x + 3)$. First, divide both sides by 2: $11=2x + 3$. Then subtract 3 from both sides: $2x=11 - 3=8$. Divide by 2: $x = 4$ cm.

Step7: Use perimeter formula for regular octagon to find $x$ in problem 5

For the regular octagon with perimeter $P = 80$ ft and side - length $x$, we use $P = 8x$. So $8x=80$, and $x = 10$ ft.

Step8: Find side - lengths and use perimeter formula to find $x$ in problem 6

The lengths of the sides of the polygon are $3$ m, $6$ m, $2$ m, $x$ m, $5$ m, and $10$ m. The perimeter $P=30$ m. So $3+6 + 2+x+5 + 10=30$. Combine like - terms: $26+x=30$. Subtract 26 from both sides: $x = 4$ m.

Answer:

1. $24$ in
2. $72$ cm
3. $26$ mm
4. $x = 4$ cm
5. $x = 10$ ft
6. $x = 4$ m