Question

1. (3 - 30) enlarge the given shape on graph paper using a zoom factor of 2. then find the perimeter and area of both shapes. what do you notice when you compare the perimeters? the areas?

Explanation:

Step1: Calculate the perimeter of the original shape

Add up the side - lengths. $P_1=6 + 5+3 + 4=18$.

Step2: Calculate the area of the original shape

Split the shape into a rectangle and a right - triangle. The rectangle has sides 3 and 4, and the right - triangle has base $4$ and height $(6 - 3)=3$. The area of the rectangle $A_{rect}=3\times4 = 12$, and the area of the triangle $A_{tri}=\frac{1}{2}\times4\times3=6$. So $A_1=12 + 6=18$.

Step3: Calculate the side - lengths of the enlarged shape

Multiply each side - length of the original shape by the zoom factor of 2. The new side - lengths are $6\times2 = 12$, $5\times2 = 10$, $3\times2 = 6$, $4\times2 = 8$.

Step4: Calculate the perimeter of the enlarged shape

$P_2=12 + 10+6 + 8=36$.

Step5: Calculate the area of the enlarged shape

The new rectangle has sides 6 and 8, and the new right - triangle has base 8 and height $(12 - 6)=6$. The area of the new rectangle $A_{new - rect}=6\times8 = 48$, and the area of the new triangle $A_{new - tri}=\frac{1}{2}\times8\times6=24$. So $A_2=48 + 24=72$.

Step6: Compare perimeters and areas

The ratio of the perimeters is $\frac{P_2}{P_1}=\frac{36}{18}=2$. The ratio of the areas is $\frac{A_2}{A_1}=\frac{72}{18}=4$. When a shape is enlarged by a factor of $k$, the perimeter is enlarged by a factor of $k$ and the area is enlarged by a factor of $k^{2}$.

Answer:

Original perimeter: 18, Original area: 18, Enlarged perimeter: 36, Enlarged area: 72. The perimeter of the enlarged shape is 2 times the original perimeter and the area of the enlarged shape is 4 times the original area.