Question
On a coordinate plane, an image rectangle has points (negative 5, 0), (5, 0), A prime (5, negative 5), (negative 5, negative 5).
An image of a dilation about the origin is shown.
Which represents the pre-image of A'?
(5, –5)
(0, 0)
(3, –3)
(Five-thirds, negative five-thirds)
An image of a dilation about the origin is shown.
Which represents the pre-image of A'?
(5, –5)
(0, 0)
(3, –3)
(Five-thirds, negative five-thirds)
Answers
GPT-4o mini
To determine the pre-image of the point A' whose coordinates are (5, -5), we need to understand how dilation works. A dilation about the origin means that the points are scaled by a certain factor from the origin (0, 0).
If we denote the dilation factor by \( k \), and the original point (pre-image) is given by coordinates \( (x, y) \), then the coordinates after dilation (the image) are given by:
\[
(x', y') = (k \cdot x, k \cdot y)
\]
For the point \( A' \) with coordinates (5, -5), we can express this as:
\[
(5, -5) = (k \cdot x, k \cdot y)
\]
To find the pre-image, we need to assume a dilation factor \( k \). The choices for the pre-image given are:
1. (5, -5)
2. (0, 0)
3. (3, -3)
4. (\( \frac{5}{3}, \frac{-5}{3} \))
If we consider a dilation that scales the original rectangle, we can check each option to see if it can result in (5, -5) through a dilation.
1. For (5, -5) as the pre-image, if \( k \) = 1, then its image would be itself. So, it cannot be the pre-image unless there's no dilation.
2. For (0, 0), the image would remain at the origin, which does not correspond to (5, -5).
3. For (3, -3), assuming a dilation of \( k \approx 1.67 \),
\[
k \cdot (3, -3) = (5, -5) \text{ when } k = \frac{5}{3} \text{, valid.}
\]
4. For \( \left(\frac{5}{3}, \frac{-5}{3}\right) \), the dilation would be:
\[
k \cdot \left(\frac{5}{3}, \frac{-5}{3}\right) = (5, -5) \text{ when } k = 3, \text{ valid.}
\]
Since both (3, -3) and \( \left(\frac{5}{3}, \frac{-5}{3}\right) \) can yield (5, -5) under dilation, confirming that both could be considered valid options.
However, to strictly answer your question based on proportionality and within the context of choosing one answer that represents the most directly scaled point, we find:
**The most straightforward pre-image of A' corresponds to (3, -3).**
So the anti-dilated coordinate (pre-image) of point A' is **(3, -3)**.
If we denote the dilation factor by \( k \), and the original point (pre-image) is given by coordinates \( (x, y) \), then the coordinates after dilation (the image) are given by:
\[
(x', y') = (k \cdot x, k \cdot y)
\]
For the point \( A' \) with coordinates (5, -5), we can express this as:
\[
(5, -5) = (k \cdot x, k \cdot y)
\]
To find the pre-image, we need to assume a dilation factor \( k \). The choices for the pre-image given are:
1. (5, -5)
2. (0, 0)
3. (3, -3)
4. (\( \frac{5}{3}, \frac{-5}{3} \))
If we consider a dilation that scales the original rectangle, we can check each option to see if it can result in (5, -5) through a dilation.
1. For (5, -5) as the pre-image, if \( k \) = 1, then its image would be itself. So, it cannot be the pre-image unless there's no dilation.
2. For (0, 0), the image would remain at the origin, which does not correspond to (5, -5).
3. For (3, -3), assuming a dilation of \( k \approx 1.67 \),
\[
k \cdot (3, -3) = (5, -5) \text{ when } k = \frac{5}{3} \text{, valid.}
\]
4. For \( \left(\frac{5}{3}, \frac{-5}{3}\right) \), the dilation would be:
\[
k \cdot \left(\frac{5}{3}, \frac{-5}{3}\right) = (5, -5) \text{ when } k = 3, \text{ valid.}
\]
Since both (3, -3) and \( \left(\frac{5}{3}, \frac{-5}{3}\right) \) can yield (5, -5) under dilation, confirming that both could be considered valid options.
However, to strictly answer your question based on proportionality and within the context of choosing one answer that represents the most directly scaled point, we find:
**The most straightforward pre-image of A' corresponds to (3, -3).**
So the anti-dilated coordinate (pre-image) of point A' is **(3, -3)**.