Question

two picture frames are shown that are similar in shape.
10 in. 6 in. 20 in. x=? in.
x = \square in.
the perimeter of the smaller frame = \square in.
the perimeter of the larger frame = \square in.

Explanation:

Step1: Set up proportion for similar rectangles

Since the frames are similar, the ratios of corresponding sides are equal. So, $\frac{10}{20} = \frac{6}{x}$.

Step2: Solve for \( x \)

Cross - multiply: \( 10x = 20\times6 \).
Simplify the right - hand side: \( 10x = 120 \).
Divide both sides by 10: \( x=\frac{120}{10}=12 \).

Step3: Calculate perimeter of smaller frame

The formula for the perimeter of a rectangle is \( P = 2\times(\text{length}+\text{width}) \). For the smaller frame, length = 10 in and width = 6 in. So, \( P_{small}=2\times(10 + 6)=2\times16 = 32 \) in.

Step4: Calculate perimeter of larger frame

For the larger frame, length = 20 in and width = 12 in. Using the perimeter formula, \( P_{large}=2\times(20 + 12)=2\times32 = 64 \) in.

Answer:

\( x = 12 \) in.
The perimeter of the smaller frame = 32 in.
The perimeter of the larger frame = 64 in.