Question
Determine the scale factor use din the dilation of line segement AB centered at the origin A(1, -3) B(2, -4)
2
4
1/4
-4
2
4
1/4
-4
Answers
GPT-4o mini
To determine the scale factor used in the dilation of line segment AB centered at the origin, we need to find the original distance between points A and B and the distance after dilation.
1. **Find the distance between points A and B:**
The coordinates of points A and B are:
- A(1, -3)
- B(2, -4)
We can use the distance formula to find the distance \(d\) between points A and B:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Plugging in the coordinates:
\[
d = \sqrt{(2 - 1)^2 + (-4 + 3)^2}
\]
\[
= \sqrt{(1)^2 + (-1)^2}
\]
\[
= \sqrt{1 + 1}
\]
\[
= \sqrt{2}
\]
2. **Assuming a dilation from the origin:**
After dilation by a scale factor \(k\), the new coordinates of A and B (denoted as \(A'\) and \(B'\)) will be:
- \(A'(k \cdot 1, k \cdot -3)\) → \(A'(k, -3k)\)
- \(B'(k \cdot 2, k \cdot -4)\) → \(B'(2k, -4k)\)
3. **Calculate the distance after dilation:**
Using the new coordinates:
\[
d' = \sqrt{(2k - k)^2 + (-4k + 3k)^2}
\]
\[
= \sqrt{(k)^2 + (-k)^2}
\]
\[
= \sqrt{k^2 + k^2}
\]
\[
= \sqrt{2k^2}
\]
\[
= k\sqrt{2}
\]
4. **Determine the scale factor:**
The scale factor \(k\) can be found by comparing the new distance \(d'\) to the original distance \(d\):
\[
\frac{d'}{d} = \frac{k\sqrt{2}}{\sqrt{2}} = k
\]
Given the options provided (2, 4, 1/4, -4), we don't have a specified new distance or a specific value for \(k\).
But knowing that dilation factors are generally positive and if \(k\) is a valid dilation factor, its value must be positive.
As no additional information is provided to determine a specific dilation or if it denotes a specific context where it must conform to one of the given options, a conclusive scale factor is not derivable from the provided information.
If we had a specific new length after dilation or additional parameters, we could solve for \(k\).
Based on typical choice constraints, you would typically select 2 or 4 as plausible options, assuming some standard dilation occurs. However, without specific criteria, I suggest validating against additional context. In dilation, the factor is usually positive and increasing distances should be observed.
**Conclusion**: More information is required to determine a specific scale factor from the provided options.
1. **Find the distance between points A and B:**
The coordinates of points A and B are:
- A(1, -3)
- B(2, -4)
We can use the distance formula to find the distance \(d\) between points A and B:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Plugging in the coordinates:
\[
d = \sqrt{(2 - 1)^2 + (-4 + 3)^2}
\]
\[
= \sqrt{(1)^2 + (-1)^2}
\]
\[
= \sqrt{1 + 1}
\]
\[
= \sqrt{2}
\]
2. **Assuming a dilation from the origin:**
After dilation by a scale factor \(k\), the new coordinates of A and B (denoted as \(A'\) and \(B'\)) will be:
- \(A'(k \cdot 1, k \cdot -3)\) → \(A'(k, -3k)\)
- \(B'(k \cdot 2, k \cdot -4)\) → \(B'(2k, -4k)\)
3. **Calculate the distance after dilation:**
Using the new coordinates:
\[
d' = \sqrt{(2k - k)^2 + (-4k + 3k)^2}
\]
\[
= \sqrt{(k)^2 + (-k)^2}
\]
\[
= \sqrt{k^2 + k^2}
\]
\[
= \sqrt{2k^2}
\]
\[
= k\sqrt{2}
\]
4. **Determine the scale factor:**
The scale factor \(k\) can be found by comparing the new distance \(d'\) to the original distance \(d\):
\[
\frac{d'}{d} = \frac{k\sqrt{2}}{\sqrt{2}} = k
\]
Given the options provided (2, 4, 1/4, -4), we don't have a specified new distance or a specific value for \(k\).
But knowing that dilation factors are generally positive and if \(k\) is a valid dilation factor, its value must be positive.
As no additional information is provided to determine a specific dilation or if it denotes a specific context where it must conform to one of the given options, a conclusive scale factor is not derivable from the provided information.
If we had a specific new length after dilation or additional parameters, we could solve for \(k\).
Based on typical choice constraints, you would typically select 2 or 4 as plausible options, assuming some standard dilation occurs. However, without specific criteria, I suggest validating against additional context. In dilation, the factor is usually positive and increasing distances should be observed.
**Conclusion**: More information is required to determine a specific scale factor from the provided options.