Question
Carly stated, “All pairs of rectangles are dilations.” Which pair of rectangles would prove that Carly’s statement is incorrect?
A rectangle with length 4 and width 2. A rectangle with length 8 and width 4.
A rectangle with length 4 and width 2. A rectangle with length 6 and width 3.
A rectangle with length 4 and width 3. A rectangle with length 3 and width 2.
A rectangle with length 4 and width 3. A rectangle with length 2 and width 1.5.
A rectangle with length 4 and width 2. A rectangle with length 8 and width 4.
A rectangle with length 4 and width 2. A rectangle with length 6 and width 3.
A rectangle with length 4 and width 3. A rectangle with length 3 and width 2.
A rectangle with length 4 and width 3. A rectangle with length 2 and width 1.5.
Answers
GPT-4o mini
To prove Carly's statement incorrect, we need to identify a pair of rectangles that are not dilations of each other.
A dilation is a transformation that alters the size of a shape but preserves its proportions. This means that the ratio of corresponding sides must be the same for both rectangles in a dilation.
Let's analyze each pair:
1. **Rectangle with length 4 and width 2** and **rectangle with length 8 and width 4**:
- For the first rectangle, the ratio of length to width = \( \frac{4}{2} = 2 \).
- For the second rectangle, the ratio of length to width = \( \frac{8}{4} = 2 \).
- These rectangles have the same ratio, so they are dilations.
2. **Rectangle with length 4 and width 2** and **rectangle with length 6 and width 3**:
- First rectangle ratio = \( \frac{4}{2} = 2 \).
- Second rectangle ratio = \( \frac{6}{3} = 2 \).
- They have the same ratio, so they are dilations.
3. **Rectangle with length 4 and width 3** and **rectangle with length 3 and width 2**:
- First rectangle ratio = \( \frac{4}{3} \).
- Second rectangle ratio = \( \frac{3}{2} \).
- The ratios are not the same (\( \frac{4}{3} \neq \frac{3}{2} \)), thus they are not dilations.
4. **Rectangle with length 4 and width 3** and **rectangle with length 2 and width 1.5**:
- First rectangle ratio = \( \frac{4}{3} \).
- Second rectangle ratio = \( \frac{2}{1.5} = \frac{4}{3} \).
- Both have the same ratio, so they are dilations.
Based on this analysis, the pair of rectangles that proves Carly's statement incorrect is:
**A rectangle with length 4 and width 3 and a rectangle with length 3 and width 2.**
These rectangles are not dilations of each other due to their differing ratios.
A dilation is a transformation that alters the size of a shape but preserves its proportions. This means that the ratio of corresponding sides must be the same for both rectangles in a dilation.
Let's analyze each pair:
1. **Rectangle with length 4 and width 2** and **rectangle with length 8 and width 4**:
- For the first rectangle, the ratio of length to width = \( \frac{4}{2} = 2 \).
- For the second rectangle, the ratio of length to width = \( \frac{8}{4} = 2 \).
- These rectangles have the same ratio, so they are dilations.
2. **Rectangle with length 4 and width 2** and **rectangle with length 6 and width 3**:
- First rectangle ratio = \( \frac{4}{2} = 2 \).
- Second rectangle ratio = \( \frac{6}{3} = 2 \).
- They have the same ratio, so they are dilations.
3. **Rectangle with length 4 and width 3** and **rectangle with length 3 and width 2**:
- First rectangle ratio = \( \frac{4}{3} \).
- Second rectangle ratio = \( \frac{3}{2} \).
- The ratios are not the same (\( \frac{4}{3} \neq \frac{3}{2} \)), thus they are not dilations.
4. **Rectangle with length 4 and width 3** and **rectangle with length 2 and width 1.5**:
- First rectangle ratio = \( \frac{4}{3} \).
- Second rectangle ratio = \( \frac{2}{1.5} = \frac{4}{3} \).
- Both have the same ratio, so they are dilations.
Based on this analysis, the pair of rectangles that proves Carly's statement incorrect is:
**A rectangle with length 4 and width 3 and a rectangle with length 3 and width 2.**
These rectangles are not dilations of each other due to their differing ratios.