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Sagot :
To solve this problem, we need to deduce the coordinates of the pre-image [tex]$\overline{Y Z}$[/tex] of the given image [tex]$\overline{Y^{\prime} Z^{\prime}}$[/tex], which has endpoints at [tex]$Y(0,3)$[/tex] and [tex]$Z(-6,3)$[/tex], after a dilation with a scale factor of 3 from the origin.
Let's break down the steps in solving this problem:
1. Understanding Dilation: Dilation is a transformation that changes the size of a figure. The scale factor tells us how much larger or smaller the image will be compared to the pre-image. Here, a scale factor of 3 means that the image [tex]$\overline{Y^{\prime} Z^{\prime}}$[/tex] is three times the size of the pre-image [tex]$\overline{Y Z}$[/tex].
2. Finding the Pre-image Coordinates:
- For dilation with a scale factor [tex]$k$[/tex], coordinates of the image are obtained by multiplying the coordinates of the pre-image by [tex]$k$[/tex]. Conversely, to find the pre-image coordinates from the image, we divide the coordinates of the image by the scale factor.
- Given points:
- [tex]$Y'(0,3)$[/tex] (image coordinate)
- [tex]$Z'(-6,3)$[/tex] (image coordinate)
- Scale factor: [tex]$3$[/tex]
To find the pre-image coordinates:
[tex]\[ \text{Pre-image } Y = \left( \frac{0}{3}, \frac{3}{3} \right) = (0.0, 1.0) \][/tex]
[tex]\[ \text{Pre-image } Z = \left( \frac{-6}{3}, \frac{3}{3} \right) = (-2.0, 1.0) \][/tex]
3. Verification:
- We calculated that the pre-image coordinates are [tex]$Y(0.0, 1.0)$[/tex] and [tex]$Z(-2.0, 1.0)$[/tex]. Now, comparing these coordinates with the given statements:
a) "[tex]$\overline{Y Z}$[/tex] is located at [tex]$Y(0,9)$[/tex] and [tex]$Z(-18,9)$[/tex] and is three times the size of [tex]$\overline{Y^{\prime} Z^{\prime}}$[/tex]" - This is not correct.
b) "[tex]$\overline{Y Z}$[/tex] is located at [tex]$Y(0,3)$[/tex] and [tex]$Z(-6,3)$[/tex] and is the same size as [tex]$\overline{Y^{\prime} Z^{\prime}}$[/tex]" - This is not correct.
c) "[tex]$[/tex]\overline{Y Z}[tex]$ is located at $[/tex]Y(0,1.5)[tex]$ and $[/tex]Z(-3,1.5)[tex]$ and is one-half the size of $[/tex]\overline{Y^{\prime} Z^{\prime}}[tex]$" - This is not correct. d) "$[/tex]\overline{Y Z}[tex]$ is located at $[/tex]Y(0,1)[tex]$ and $[/tex]Z(-2,1)[tex]$ and is one-third the size of $[/tex]\overline{Y^{\prime} Z^{\prime}}[tex]$" - This is correct. Therefore, the correct statement describing the pre-image $[/tex]\overline{Y Z}[tex]$ is: $[/tex]\quad \overline{Y Z} \text{ is located at } Y(0,1) \text{ and } Z(-2,1) \text{ and is one-third the size of } \overline{Y^{\prime} Z^{\prime}}.$
Let's break down the steps in solving this problem:
1. Understanding Dilation: Dilation is a transformation that changes the size of a figure. The scale factor tells us how much larger or smaller the image will be compared to the pre-image. Here, a scale factor of 3 means that the image [tex]$\overline{Y^{\prime} Z^{\prime}}$[/tex] is three times the size of the pre-image [tex]$\overline{Y Z}$[/tex].
2. Finding the Pre-image Coordinates:
- For dilation with a scale factor [tex]$k$[/tex], coordinates of the image are obtained by multiplying the coordinates of the pre-image by [tex]$k$[/tex]. Conversely, to find the pre-image coordinates from the image, we divide the coordinates of the image by the scale factor.
- Given points:
- [tex]$Y'(0,3)$[/tex] (image coordinate)
- [tex]$Z'(-6,3)$[/tex] (image coordinate)
- Scale factor: [tex]$3$[/tex]
To find the pre-image coordinates:
[tex]\[ \text{Pre-image } Y = \left( \frac{0}{3}, \frac{3}{3} \right) = (0.0, 1.0) \][/tex]
[tex]\[ \text{Pre-image } Z = \left( \frac{-6}{3}, \frac{3}{3} \right) = (-2.0, 1.0) \][/tex]
3. Verification:
- We calculated that the pre-image coordinates are [tex]$Y(0.0, 1.0)$[/tex] and [tex]$Z(-2.0, 1.0)$[/tex]. Now, comparing these coordinates with the given statements:
a) "[tex]$\overline{Y Z}$[/tex] is located at [tex]$Y(0,9)$[/tex] and [tex]$Z(-18,9)$[/tex] and is three times the size of [tex]$\overline{Y^{\prime} Z^{\prime}}$[/tex]" - This is not correct.
b) "[tex]$\overline{Y Z}$[/tex] is located at [tex]$Y(0,3)$[/tex] and [tex]$Z(-6,3)$[/tex] and is the same size as [tex]$\overline{Y^{\prime} Z^{\prime}}$[/tex]" - This is not correct.
c) "[tex]$[/tex]\overline{Y Z}[tex]$ is located at $[/tex]Y(0,1.5)[tex]$ and $[/tex]Z(-3,1.5)[tex]$ and is one-half the size of $[/tex]\overline{Y^{\prime} Z^{\prime}}[tex]$" - This is not correct. d) "$[/tex]\overline{Y Z}[tex]$ is located at $[/tex]Y(0,1)[tex]$ and $[/tex]Z(-2,1)[tex]$ and is one-third the size of $[/tex]\overline{Y^{\prime} Z^{\prime}}[tex]$" - This is correct. Therefore, the correct statement describing the pre-image $[/tex]\overline{Y Z}[tex]$ is: $[/tex]\quad \overline{Y Z} \text{ is located at } Y(0,1) \text{ and } Z(-2,1) \text{ and is one-third the size of } \overline{Y^{\prime} Z^{\prime}}.$
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