Question
On a coordinate plane, a line segment contains points (negative 5, 5) and (5, 0). Another segment contains points (negative 1, 1) and (1, 0).
The segments shown are dilations of each other about the origin. Which statement could be true?
The coordinate (1, 0) is from a dilation using the scale factor of One-fifth.
The coordinate (5, 0) is from a dilation using the scale factor of 4.
The coordinate (1, 0) is from a dilation using the scale factor of 5.
The coordinate (5, 0) is from a dilation using the scale factor of One-fifth.
The segments shown are dilations of each other about the origin. Which statement could be true?
The coordinate (1, 0) is from a dilation using the scale factor of One-fifth.
The coordinate (5, 0) is from a dilation using the scale factor of 4.
The coordinate (1, 0) is from a dilation using the scale factor of 5.
The coordinate (5, 0) is from a dilation using the scale factor of One-fifth.
Answers
GPT-4o mini
To determine which statement could be true regarding the dilation of one line segment to another, we first need to confirm the lengths and ratios of the two segments.
**First segment endpoints:**
- \( A(-5, 5) \)
- \( B(5, 0) \)
**Length of first segment:**
\[
\text{Length} = \sqrt{(5 - (-5))^2 + (0 - 5)^2} = \sqrt{(5 + 5)^2 + (0 - 5)^2} = \sqrt{10^2 + (-5)^2} = \sqrt{100 + 25} = \sqrt{125} = 5\sqrt{5}
\]
**Second segment endpoints:**
- \( C(-1, 1) \)
- \( D(1, 0) \)
**Length of second segment:**
\[
\text{Length} = \sqrt{(1 - (-1))^2 + (0 - 1)^2} = \sqrt{(1 + 1)^2 + (0 - 1)^2} = \sqrt{2^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}
\]
**Finding the scale factor between the segments:**
To find the scale factor \( k \), we compare the lengths of the segments:
\[
k = \frac{\text{Length of first segment}}{\text{Length of second segment}} = \frac{5\sqrt{5}}{\sqrt{5}} = 5
\]
Now, we will evaluate the statements given:
1. **The coordinate (1, 0) is from a dilation using the scale factor of One-fifth.**
- This is incorrect since we found the scale factor to be 5, not \( \frac{1}{5} \).
2. **The coordinate (5, 0) is from a dilation using the scale factor of 4.**
- This is incorrect, as we found the scale factor to be 5, not 4.
3. **The coordinate (1, 0) is from a dilation using the scale factor of 5.**
- This is correct since (1, 0) is a point on the smaller segment, and a dilation by a scale factor of 5 would yield (5, 0).
4. **The coordinate (5, 0) is from a dilation using the scale factor of One-fifth.**
- This is incorrect, for the same reason as above: our scale factor is 5, not \( \frac{1}{5} \).
Based on the analysis, the correct statement that could be true is:
**The coordinate (1, 0) is from a dilation using the scale factor of 5.**
**First segment endpoints:**
- \( A(-5, 5) \)
- \( B(5, 0) \)
**Length of first segment:**
\[
\text{Length} = \sqrt{(5 - (-5))^2 + (0 - 5)^2} = \sqrt{(5 + 5)^2 + (0 - 5)^2} = \sqrt{10^2 + (-5)^2} = \sqrt{100 + 25} = \sqrt{125} = 5\sqrt{5}
\]
**Second segment endpoints:**
- \( C(-1, 1) \)
- \( D(1, 0) \)
**Length of second segment:**
\[
\text{Length} = \sqrt{(1 - (-1))^2 + (0 - 1)^2} = \sqrt{(1 + 1)^2 + (0 - 1)^2} = \sqrt{2^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}
\]
**Finding the scale factor between the segments:**
To find the scale factor \( k \), we compare the lengths of the segments:
\[
k = \frac{\text{Length of first segment}}{\text{Length of second segment}} = \frac{5\sqrt{5}}{\sqrt{5}} = 5
\]
Now, we will evaluate the statements given:
1. **The coordinate (1, 0) is from a dilation using the scale factor of One-fifth.**
- This is incorrect since we found the scale factor to be 5, not \( \frac{1}{5} \).
2. **The coordinate (5, 0) is from a dilation using the scale factor of 4.**
- This is incorrect, as we found the scale factor to be 5, not 4.
3. **The coordinate (1, 0) is from a dilation using the scale factor of 5.**
- This is correct since (1, 0) is a point on the smaller segment, and a dilation by a scale factor of 5 would yield (5, 0).
4. **The coordinate (5, 0) is from a dilation using the scale factor of One-fifth.**
- This is incorrect, for the same reason as above: our scale factor is 5, not \( \frac{1}{5} \).
Based on the analysis, the correct statement that could be true is:
**The coordinate (1, 0) is from a dilation using the scale factor of 5.**