Question

1. given the following rug pattern, think about what the rug would look like if it was 12 times larger than figure #1. what would the perimeter and area of this new rug (figure 12)?

Explanation:

1. First, assume the side - length of each small square in figure #1 is \(s = 1\) unit.

• Let's analyze the perimeter and area relationships for similar figures. For two similar figures, if the scale factor of enlargement is \(k\), the ratio of the perimeters of two similar figures is equal to the scale factor, and the ratio of the areas of two similar figures is equal to \(k^{2}\).
• In figure #1, assume we can count the number of outer - side lengths of the small squares to find the perimeter. Let's assume the perimeter of figure #1 is \(P_1\) and the area is \(A_1\). If we count the outer - side lengths of the small squares in figure #1, assume \(P_1 = 8\) (by counting the number of unit - length sides on the outer boundary) and \(A_1=4\) (by counting the number of unit squares).

1. The scale factor \(k = 12\).

• Perimeter of the new rug (figure 12):
• The formula for the perimeter of a similar figure is \(P_2=k\times P_1\).
• Substitute \(k = 12\) and \(P_1 = 8\) into the formula. So \(P_2=12\times8=96\) units.
• Area of the new rug (figure 12):
• The formula for the area of a similar figure is \(A_2=k^{2}\times A_1\).
• Substitute \(k = 12\) and \(A_1 = 4\) into the formula. So \(A_2=12^{2}\times4=144\times4 = 576\) square units.

Step1: Identify scale - factor and initial perimeter/area

Let the scale factor \(k = 12\), assume initial perimeter \(P_1 = 8\) and initial area \(A_1 = 4\).

Step2: Calculate new perimeter

Use the formula \(P_2=k\times P_1\), so \(P_2=12\times8\).

Step3: Calculate new area

Use the formula \(A_2=k^{2}\times A_1\), so \(A_2 = 12^{2}\times4\).

Answer:

Perimeter: 96 units, Area: 576 square units