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Sagot :
Let’s break down the problem step by step and find the required values.
### Step 1: Calculate the Slope of Line Segment [tex]\(\overline{WX}\)[/tex]
Given the coordinates:
- Point [tex]\(W\)[/tex] is at [tex]\((3, 2)\)[/tex]
- Point [tex]\(X\)[/tex] is at [tex]\((7, 5)\)[/tex]
The formula to calculate the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the coordinates [tex]\((3, 2)\)[/tex] and [tex]\((7, 5)\)[/tex]:
[tex]\[ \text{slope} = \frac{5 - 2}{7 - 3} = \frac{3}{4} \][/tex]
So, the slope of [tex]\(\overline{WX}\)[/tex] is [tex]\(\frac{3}{4}\)[/tex].
### Step 2: Calculate the Length of Line Segment [tex]\(\overline{WX}\)[/tex]
The formula to calculate the distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ \text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Using the coordinates [tex]\((3, 2)\)[/tex] and [tex]\((7, 5)\)[/tex]:
[tex]\[ \text{distance} = \sqrt{(7 - 3)^2 + (5 - 2)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \][/tex]
So, the length of [tex]\(\overline{WX}\)[/tex] is [tex]\(5\)[/tex].
### Step 3: Dilation of the Polygon
The dilation scale factor is [tex]\(3\)[/tex]. This means that the new length of [tex]\(\overline{WX}\)[/tex] after dilation will be three times the original length:
[tex]\[ \text{New length} = 5 \times 3 = 15 \][/tex]
### Selecting the Correct Statement
Let's evaluate the given statements based on our findings:
A. The slope of [tex]\(\overline{WX}\)[/tex] is [tex]\(\frac{3}{4}\)[/tex], and the length of [tex]\(\overline{WX}\)[/tex] is [tex]\(5\)[/tex].
True: The slope is [tex]\(\frac{3}{4}\)[/tex] and the original length is [tex]\(5\)[/tex].
B. The slope of [tex]\(\overline{WX}\)[/tex] is [tex]\(\frac{3}{4}\)[/tex], and the length of [tex]\(\overline{WX}\)[/tex] is [tex]\(15\)[/tex].
False: The length after dilation is [tex]\(15\)[/tex], but the original length is [tex]\(5\)[/tex].
C. The slope of [tex]\(\overline{Wx}\)[/tex] is [tex]\(\frac{8}{4}\)[/tex], and the length of [tex]\(\overline{Wx}\)[/tex] is [tex]\(15\)[/tex].
False: The slope is incorrect (it should be [tex]\(\frac{3}{4}\)[/tex]). Also, the length mentioned is the length after dilation, not the original length.
D. The slope of [tex]\(\overline{Wx}\)[/tex] is [tex]\(\frac{9}{4}\)[/tex], and the length of [tex]\(\overline{WX}\)[/tex] is [tex]\(5\)[/tex].
False: Both the slope and length are incorrect. The slope should be [tex]\(\frac{3}{4}\)[/tex] and the original length is [tex]\(5\)[/tex].
Therefore, the correct statement is:
A. The slope of [tex]\(\overline{WX}\)[/tex] is [tex]\(\frac{3}{4}\)[/tex], and the length of [tex]\(\overline{WX}\)[/tex] is [tex]\(5\)[/tex].
### Step 1: Calculate the Slope of Line Segment [tex]\(\overline{WX}\)[/tex]
Given the coordinates:
- Point [tex]\(W\)[/tex] is at [tex]\((3, 2)\)[/tex]
- Point [tex]\(X\)[/tex] is at [tex]\((7, 5)\)[/tex]
The formula to calculate the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the coordinates [tex]\((3, 2)\)[/tex] and [tex]\((7, 5)\)[/tex]:
[tex]\[ \text{slope} = \frac{5 - 2}{7 - 3} = \frac{3}{4} \][/tex]
So, the slope of [tex]\(\overline{WX}\)[/tex] is [tex]\(\frac{3}{4}\)[/tex].
### Step 2: Calculate the Length of Line Segment [tex]\(\overline{WX}\)[/tex]
The formula to calculate the distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ \text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Using the coordinates [tex]\((3, 2)\)[/tex] and [tex]\((7, 5)\)[/tex]:
[tex]\[ \text{distance} = \sqrt{(7 - 3)^2 + (5 - 2)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \][/tex]
So, the length of [tex]\(\overline{WX}\)[/tex] is [tex]\(5\)[/tex].
### Step 3: Dilation of the Polygon
The dilation scale factor is [tex]\(3\)[/tex]. This means that the new length of [tex]\(\overline{WX}\)[/tex] after dilation will be three times the original length:
[tex]\[ \text{New length} = 5 \times 3 = 15 \][/tex]
### Selecting the Correct Statement
Let's evaluate the given statements based on our findings:
A. The slope of [tex]\(\overline{WX}\)[/tex] is [tex]\(\frac{3}{4}\)[/tex], and the length of [tex]\(\overline{WX}\)[/tex] is [tex]\(5\)[/tex].
True: The slope is [tex]\(\frac{3}{4}\)[/tex] and the original length is [tex]\(5\)[/tex].
B. The slope of [tex]\(\overline{WX}\)[/tex] is [tex]\(\frac{3}{4}\)[/tex], and the length of [tex]\(\overline{WX}\)[/tex] is [tex]\(15\)[/tex].
False: The length after dilation is [tex]\(15\)[/tex], but the original length is [tex]\(5\)[/tex].
C. The slope of [tex]\(\overline{Wx}\)[/tex] is [tex]\(\frac{8}{4}\)[/tex], and the length of [tex]\(\overline{Wx}\)[/tex] is [tex]\(15\)[/tex].
False: The slope is incorrect (it should be [tex]\(\frac{3}{4}\)[/tex]). Also, the length mentioned is the length after dilation, not the original length.
D. The slope of [tex]\(\overline{Wx}\)[/tex] is [tex]\(\frac{9}{4}\)[/tex], and the length of [tex]\(\overline{WX}\)[/tex] is [tex]\(5\)[/tex].
False: Both the slope and length are incorrect. The slope should be [tex]\(\frac{3}{4}\)[/tex] and the original length is [tex]\(5\)[/tex].
Therefore, the correct statement is:
A. The slope of [tex]\(\overline{WX}\)[/tex] is [tex]\(\frac{3}{4}\)[/tex], and the length of [tex]\(\overline{WX}\)[/tex] is [tex]\(5\)[/tex].
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