Question

a trapezoidal door mat is similar in shape to a trapezoidal rug. each dimension of the door mat is $\frac{1}{3}$ the size of the rug. which statement is true? a. the area of the door mat is $\frac{1}{6}$ the area of the rug. b. the area of the door mat is $\frac{1}{9}$ the area of the rug. c. the perimeter of the door mat is $\frac{1}{6}$ the perimeter of the rug. d. the perimeter of the door mat is $\frac{1}{9}$ the perimeter of the rug.

Explanation:

Step1: Recall ratio - of - similar - figures rules

For two similar figures, if the ratio of their corresponding side lengths is \(k\), the ratio of their perimeters is \(k\) and the ratio of their areas is \(k^{2}\). Here, \(k=\frac{1}{3}\) since each dimension of the door - mat is \(\frac{1}{3}\) the size of the rug.

Step2: Calculate the ratio of perimeters

The ratio of the perimeters of two similar figures is equal to the ratio of their corresponding side lengths. So, if the ratio of side lengths \(k = \frac{1}{3}\), the ratio of the perimeter of the door - mat to the perimeter of the rug is \(\frac{1}{3}\).

Step3: Calculate the ratio of areas

The ratio of the areas of two similar figures is \(k^{2}\). Since \(k=\frac{1}{3}\), then \(k^{2}=(\frac{1}{3})^{2}=\frac{1}{9}\). So the area of the door - mat is \(\frac{1}{9}\) the area of the rug.

Answer:

B. The area of the door mat is \(\frac{1}{9}\) the area of the rug.