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Sagot :
Let's analyze the properties of the dilation of polygon WXYZ where vertex [tex]\( W \)[/tex] is the center of dilation, resulting in polygon [tex]\( W^{\prime}X^{\prime}Y^{\prime}Z \)[/tex]. We are given that the coordinates of point [tex]\( W \)[/tex] are [tex]\( (3, 2) \)[/tex], and the coordinates of point [tex]\( X \)[/tex] are [tex]\( (7.5, 2) \)[/tex].
We need to address two properties:
1. The slope of line segment [tex]\( \overline{WX} \)[/tex].
2. The length of line segment [tex]\( \overline{WX} \)[/tex].
### 1. Slope of [tex]\( \overline{WX} \)[/tex]
The slope of a line between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute [tex]\((x_1, y_1) = (3, 2)\)[/tex] and [tex]\((x_2, y_2) = (7.5, 2)\)[/tex]:
[tex]\[ \text{slope of } \overline{WX} = \frac{2 - 2}{7.5 - 3} = \frac{0}{4.5} = 0 \][/tex]
The slope of [tex]\( \overline{WX} \)[/tex] is therefore 0.
### 2. Length of [tex]\( \overline{WX} \)[/tex]
The length of a line segment between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the distance formula:
[tex]\[ \text{length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Substitute [tex]\((x_1, y_1) = (3, 2)\)[/tex] and [tex]\((x_2, y_2) = (7.5, 2)\)[/tex]:
[tex]\[ \text{length of } \overline{WX} = \sqrt{(7.5 - 3)^2 + (2 - 2)^2} = \sqrt{4.5^2 + 0^2} = \sqrt{20.25} = 4.5 \][/tex]
The length of [tex]\( \overline{WX} \)[/tex] is 4.5.
### 3. Length of [tex]\( \overline{W'X'} \)[/tex] after dilation
Since the polygon WXYZ is dilated by a scale factor of 3 with [tex]\( W \)[/tex] as the center of dilation, the length of [tex]\( \overline{W'X'} \)[/tex] will be 3 times the original length:
[tex]\[ \text{length of } \overline{W'X'} = 3 \times 4.5 = 13.5 \][/tex]
### Conclusion
Based on this, the correct statement would involve the calculated slope (0) and the calculated lengths [tex]\( \overline{WX} = 4.5 \)[/tex] and [tex]\( \overline{W'X'} = 13.5 \)[/tex].
However, none of the provided options in the problem seems to match the correct values derived from the given data. Ensure that the options are reviewed or resubmitted to match factual mathematical analysis.
We need to address two properties:
1. The slope of line segment [tex]\( \overline{WX} \)[/tex].
2. The length of line segment [tex]\( \overline{WX} \)[/tex].
### 1. Slope of [tex]\( \overline{WX} \)[/tex]
The slope of a line between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute [tex]\((x_1, y_1) = (3, 2)\)[/tex] and [tex]\((x_2, y_2) = (7.5, 2)\)[/tex]:
[tex]\[ \text{slope of } \overline{WX} = \frac{2 - 2}{7.5 - 3} = \frac{0}{4.5} = 0 \][/tex]
The slope of [tex]\( \overline{WX} \)[/tex] is therefore 0.
### 2. Length of [tex]\( \overline{WX} \)[/tex]
The length of a line segment between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the distance formula:
[tex]\[ \text{length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Substitute [tex]\((x_1, y_1) = (3, 2)\)[/tex] and [tex]\((x_2, y_2) = (7.5, 2)\)[/tex]:
[tex]\[ \text{length of } \overline{WX} = \sqrt{(7.5 - 3)^2 + (2 - 2)^2} = \sqrt{4.5^2 + 0^2} = \sqrt{20.25} = 4.5 \][/tex]
The length of [tex]\( \overline{WX} \)[/tex] is 4.5.
### 3. Length of [tex]\( \overline{W'X'} \)[/tex] after dilation
Since the polygon WXYZ is dilated by a scale factor of 3 with [tex]\( W \)[/tex] as the center of dilation, the length of [tex]\( \overline{W'X'} \)[/tex] will be 3 times the original length:
[tex]\[ \text{length of } \overline{W'X'} = 3 \times 4.5 = 13.5 \][/tex]
### Conclusion
Based on this, the correct statement would involve the calculated slope (0) and the calculated lengths [tex]\( \overline{WX} = 4.5 \)[/tex] and [tex]\( \overline{W'X'} = 13.5 \)[/tex].
However, none of the provided options in the problem seems to match the correct values derived from the given data. Ensure that the options are reviewed or resubmitted to match factual mathematical analysis.
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