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Sagot :
To solve the problem, we need to determine the correct statement describing the pre-image segment [tex]\( \overline{YZ} \)[/tex] after a dilation transformation from the segment [tex]\( \overline{Y^{\prime} Z^{\prime}} \)[/tex].
1. Calculate the length of [tex]\( \overline{Y^{\prime} Z^{\prime}} \)[/tex]:
The endpoints of [tex]\( \overline{Y^{\prime} Z^{\prime}} \)[/tex] are given as [tex]\( Y^{\prime}(0, 3) \)[/tex] and [tex]\( Z^{\prime}(-6, 3) \)[/tex].
The length of [tex]\( \overline{Y^{\prime} Z^{\prime}} \)[/tex] is calculated using the distance formula:
[tex]\[ \text{Length of } \overline{Y^{\prime} Z^{\prime}} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Plugging in the coordinates [tex]\( Y^{\prime}(0, 3) \)[/tex] and [tex]\( Z^{\prime}(-6, 3) \)[/tex]:
[tex]\[ \text{Length of } \overline{Y^{\prime} Z^{\prime}} = \sqrt{(-6 - 0)^2 + (3 - 3)^2} = \sqrt{(-6)^2 + 0^2} = \sqrt{36} = 6 \][/tex]
Therefore, the length of [tex]\( \overline{Y^{\prime} Z^{\prime}} \)[/tex] is [tex]\( 6 \)[/tex] units.
2. Determine the dilation transformation:
The problem states that [tex]\( \overline{YZ} \)[/tex] was dilated by a scale factor of [tex]\( 3 \)[/tex] from the origin.
Therefore, if the length of [tex]\( \overline{Y^{\prime} Z^{\prime}} \)[/tex] is [tex]\( 6 \)[/tex], then the length of [tex]\( \overline{YZ} \)[/tex] after dilation will be:
[tex]\[ \text{Length of } \overline{YZ} = 6 \times 3 = 18 \][/tex]
3. Verify the coordinates of [tex]\( \overline{YZ} \)[/tex]:
- Option 1: [tex]\( \overline{YZ} \)[/tex] is located at [tex]\( Y(0, 9) \)[/tex] and [tex]\( Z(-18, 9) \)[/tex]
Length: [tex]\( \sqrt{(-18 - 0)^2 + (9 - 9)^2} = \sqrt{(-18)^2 + 0^2} = \sqrt{324} = 18 \)[/tex]
This matches the calculated length of [tex]\( 18 \)[/tex] and also accurately reflects that [tex]\( \overline{YZ} \)[/tex] is three times the size of [tex]\( \overline{Y^{\prime} Z^{\prime}} \)[/tex].
- Other options:
- Option 2: [tex]\( Y(0, 3) \)[/tex] and [tex]\( Z(-6, 3) \)[/tex]
This matches [tex]\( Y^{\prime} \)[/tex] and [tex]\( Z^{\prime} \)[/tex], so it would be the same size as [tex]\( \overline{Y^{\prime} Z^{\prime}} \)[/tex], not three times.
- Option 3: [tex]\( Y(0, 1.5) \)[/tex] and [tex]\( Z(-3, 1.5) \)[/tex]
Length: [tex]\( \sqrt{(-3 - 0)^2 + (1.5 - 1.5)^2} = 3 \)[/tex], which is one-half, not three times.
- Option 4: [tex]\( Y(0, 1) \)[/tex] and [tex]\( Z(-2, 1) \)[/tex]
Length: [tex]\( \sqrt{(-2 - 0)^2 + (1 - 1)^2} = 2 \)[/tex], which is one-third, not three times.
Therefore, the correct statement is:
[tex]\[ \overline{YZ} \text{ is located at } Y(0, 9) \text{ and } Z(-18, 9) \text{ and is three times the size of } \overline{Y^{\prime} Z^{\prime}} \][/tex]
So the correct answer is option 1.
1. Calculate the length of [tex]\( \overline{Y^{\prime} Z^{\prime}} \)[/tex]:
The endpoints of [tex]\( \overline{Y^{\prime} Z^{\prime}} \)[/tex] are given as [tex]\( Y^{\prime}(0, 3) \)[/tex] and [tex]\( Z^{\prime}(-6, 3) \)[/tex].
The length of [tex]\( \overline{Y^{\prime} Z^{\prime}} \)[/tex] is calculated using the distance formula:
[tex]\[ \text{Length of } \overline{Y^{\prime} Z^{\prime}} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Plugging in the coordinates [tex]\( Y^{\prime}(0, 3) \)[/tex] and [tex]\( Z^{\prime}(-6, 3) \)[/tex]:
[tex]\[ \text{Length of } \overline{Y^{\prime} Z^{\prime}} = \sqrt{(-6 - 0)^2 + (3 - 3)^2} = \sqrt{(-6)^2 + 0^2} = \sqrt{36} = 6 \][/tex]
Therefore, the length of [tex]\( \overline{Y^{\prime} Z^{\prime}} \)[/tex] is [tex]\( 6 \)[/tex] units.
2. Determine the dilation transformation:
The problem states that [tex]\( \overline{YZ} \)[/tex] was dilated by a scale factor of [tex]\( 3 \)[/tex] from the origin.
Therefore, if the length of [tex]\( \overline{Y^{\prime} Z^{\prime}} \)[/tex] is [tex]\( 6 \)[/tex], then the length of [tex]\( \overline{YZ} \)[/tex] after dilation will be:
[tex]\[ \text{Length of } \overline{YZ} = 6 \times 3 = 18 \][/tex]
3. Verify the coordinates of [tex]\( \overline{YZ} \)[/tex]:
- Option 1: [tex]\( \overline{YZ} \)[/tex] is located at [tex]\( Y(0, 9) \)[/tex] and [tex]\( Z(-18, 9) \)[/tex]
Length: [tex]\( \sqrt{(-18 - 0)^2 + (9 - 9)^2} = \sqrt{(-18)^2 + 0^2} = \sqrt{324} = 18 \)[/tex]
This matches the calculated length of [tex]\( 18 \)[/tex] and also accurately reflects that [tex]\( \overline{YZ} \)[/tex] is three times the size of [tex]\( \overline{Y^{\prime} Z^{\prime}} \)[/tex].
- Other options:
- Option 2: [tex]\( Y(0, 3) \)[/tex] and [tex]\( Z(-6, 3) \)[/tex]
This matches [tex]\( Y^{\prime} \)[/tex] and [tex]\( Z^{\prime} \)[/tex], so it would be the same size as [tex]\( \overline{Y^{\prime} Z^{\prime}} \)[/tex], not three times.
- Option 3: [tex]\( Y(0, 1.5) \)[/tex] and [tex]\( Z(-3, 1.5) \)[/tex]
Length: [tex]\( \sqrt{(-3 - 0)^2 + (1.5 - 1.5)^2} = 3 \)[/tex], which is one-half, not three times.
- Option 4: [tex]\( Y(0, 1) \)[/tex] and [tex]\( Z(-2, 1) \)[/tex]
Length: [tex]\( \sqrt{(-2 - 0)^2 + (1 - 1)^2} = 2 \)[/tex], which is one-third, not three times.
Therefore, the correct statement is:
[tex]\[ \overline{YZ} \text{ is located at } Y(0, 9) \text{ and } Z(-18, 9) \text{ and is three times the size of } \overline{Y^{\prime} Z^{\prime}} \][/tex]
So the correct answer is option 1.
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