Question

In table tennis, the table is a rectangle 9 feet long and 5 feet wide. A tennis court is a rectangle 78 feet long and 36 feet wide. Are the two surfaces similar? Explain. If so, find the scale factor of the tennis court to the table.

Answer 1

To determine if the two surfaces (the table tennis table and the tennis court) are similar, we need to compare their aspect ratios. Two rectangles are similar if their corresponding side lengths are in the same ratio.

1. **Calculate the aspect ratio of the table tennis table**:
- Length of the table: 9 feet
- Width of the table: 5 feet
- Aspect ratio of the table: \( \frac{\text{Length}}{\text{Width}} = \frac{9}{5} \)

2. **Calculate the aspect ratio of the tennis court**:
- Length of the court: 78 feet
- Width of the court: 36 feet
- Aspect ratio of the court: \( \frac{\text{Length}}{\text{Width}} = \frac{78}{36} \)

To simplify \( \frac{78}{36} \):
- Find the greatest common divisor (GCD) of 78 and 36. The GCD is 6.
- Divide both numbers by their GCD:
\[
\frac{78 \div 6}{36 \div 6} = \frac{13}{6}
\]

3. **Compare the aspect ratios**:
- Aspect ratio of the table: \( \frac{9}{5} \)
- Aspect ratio of the court: \( \frac{13}{6} \)

To check if these ratios are the same, we can cross-multiply:
\[
9 \times 6 = 54 \quad \text{and} \quad 5 \times 13 = 65
\]
Since \( 54 \neq 65 \), the aspect ratios are not the same.

Since the aspect ratios of the table tennis table and the tennis court are different, the two surfaces are **not similar**.

Therefore, we conclude that the table tennis table and the tennis court are not similar surfaces.