Question

1. use the diagram of two similar rectangles below to answer the following questions.

a
b
4 ft
21 ft
7 ft
7 ft
7 ft
12 ft
4 ft
4 ft
a) what is the scale factor from rectangle a to rectangle b? 3
b) what is the scale factor from rectangle b to rectangle a? 1/3
c) calculate the area and perimeter for rectangle a:
area: 28 ft²
perimeter: 22 ft
7×4 = 28 ft²
7 + 7 = 14
4 + 4 = 8
d) calculate the area and perimeter for rectangle b:
area: 252 ft²
perimeter: 66 ft
21×12 = 252 ft²
21 + 21 = 42
12 + 12 = 24
66
e) how did the perimeter change from rectangle a to rectangle b?
f) how did the area change from rectangle a to rectangle b? 9 times

Explanation:

Step1: Calculate scale - factor from A to B

For similar rectangles, if we compare corresponding sides. Let's take the length of rectangle A ($l_A = 7$ ft) and rectangle B ($l_B=21$ ft). The scale - factor $k_{A
ightarrow B}=\frac{l_B}{l_A}=\frac{21}{7} = 3$.

Step2: Calculate scale - factor from B to A

The scale - factor $k_{B
ightarrow A}=\frac{l_A}{l_B}=\frac{7}{21}=\frac{1}{3}$.

Step3: Calculate area and perimeter of Rectangle A

The area of a rectangle is $A = l\times w$. For rectangle A with $l = 7$ ft and $w = 4$ ft, $A_A=7\times4 = 28$ ft². The perimeter of a rectangle is $P=2(l + w)$, so $P_A=2(7 + 4)=2\times11 = 22$ ft.

Step4: Calculate area and perimeter of Rectangle B

For rectangle B with $l = 21$ ft and $w = 12$ ft, the area $A_B=21\times12 = 252$ ft². The perimeter $P_B=2(21 + 12)=2\times33 = 66$ ft.

Step5: Analyze perimeter change

The ratio of the perimeters $\frac{P_B}{P_A}=\frac{66}{22}=3$. So the perimeter of rectangle B is 3 times the perimeter of rectangle A.

Step6: Analyze area change

The ratio of the areas $\frac{A_B}{A_A}=\frac{252}{28}=9$. So the area of rectangle B is 9 times the area of rectangle A.

Answer:

a) 3
b) $\frac{1}{3}$
c) Area: 28 ft², Perimeter: 22 ft
d) Area: 252 ft², Perimeter: 66 ft
e) The perimeter of Rectangle B is 3 times the perimeter of Rectangle A.
f) The area of Rectangle B is 9 times the area of Rectangle A.