Answer :
To find the correct statement among the given choices, we'll first determine the slope and the length of the line segment [tex]\(\overline{WX}\)[/tex], and then we'll understand how the dilation affects these quantities.
### Step 1: Compute the Slope of [tex]\(\overline{WX}\)[/tex]
To find the slope of the line segment [tex]\(\overline{WX}\)[/tex] which connects points [tex]\( W(3, 2) \)[/tex] and [tex]\( X(7, 5) \)[/tex], we use the slope formula:
[tex]\[ \text{slope of } \overline{WX} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in our coordinates [tex]\(W(3, 2)\)[/tex] (i.e., [tex]\( x_1 = 3 \)[/tex] and [tex]\( y_1 = 2 \)[/tex]) and [tex]\(X(7, 5)\)[/tex] (i.e., [tex]\( x_2 = 7 \)[/tex] and [tex]\( y_2 = 5 \)[/tex]), we get:
[tex]\[ \frac{5 - 2}{7 - 3} = \frac{3}{4} \][/tex]
### Step 2: Compute the Length of [tex]\(\overline{WX}\)[/tex]
To find the length of the segment [tex]\(\overline{WX}\)[/tex], we use the distance formula:
[tex]\[ \text{length of } \overline{WX} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Substituting the coordinates [tex]\(W(3, 2)\)[/tex] and [tex]\(X(7, 5)\)[/tex], we get:
[tex]\[ \sqrt{(7 - 3)^2 + (5 - 2)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \][/tex]
### Step 3: Determine the Length After Dilation
Since the polygon is dilated by a scale factor of 3 with [tex]\( W(3, 2) \)[/tex] as the center of dilation, the length of [tex]\(\overline{WX}\)[/tex] after dilation will be:
[tex]\[ 5 \times 3 = 15 \][/tex]
So, the slope of [tex]\(\overline{WX}\)[/tex] is [tex]\(\frac{3}{4}\)[/tex], and the original length of [tex]\(\overline{WX}\)[/tex] is 5. After dilation, the length is 15.
### Conclusion
The correct statement is:
C. The slope of [tex]\(\overline{W X}\)[/tex] is [tex]\(\frac{3}{4}\)[/tex], and the length of [tex]\(\overline{W X}\)[/tex] is 5.
### Step 1: Compute the Slope of [tex]\(\overline{WX}\)[/tex]
To find the slope of the line segment [tex]\(\overline{WX}\)[/tex] which connects points [tex]\( W(3, 2) \)[/tex] and [tex]\( X(7, 5) \)[/tex], we use the slope formula:
[tex]\[ \text{slope of } \overline{WX} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in our coordinates [tex]\(W(3, 2)\)[/tex] (i.e., [tex]\( x_1 = 3 \)[/tex] and [tex]\( y_1 = 2 \)[/tex]) and [tex]\(X(7, 5)\)[/tex] (i.e., [tex]\( x_2 = 7 \)[/tex] and [tex]\( y_2 = 5 \)[/tex]), we get:
[tex]\[ \frac{5 - 2}{7 - 3} = \frac{3}{4} \][/tex]
### Step 2: Compute the Length of [tex]\(\overline{WX}\)[/tex]
To find the length of the segment [tex]\(\overline{WX}\)[/tex], we use the distance formula:
[tex]\[ \text{length of } \overline{WX} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Substituting the coordinates [tex]\(W(3, 2)\)[/tex] and [tex]\(X(7, 5)\)[/tex], we get:
[tex]\[ \sqrt{(7 - 3)^2 + (5 - 2)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \][/tex]
### Step 3: Determine the Length After Dilation
Since the polygon is dilated by a scale factor of 3 with [tex]\( W(3, 2) \)[/tex] as the center of dilation, the length of [tex]\(\overline{WX}\)[/tex] after dilation will be:
[tex]\[ 5 \times 3 = 15 \][/tex]
So, the slope of [tex]\(\overline{WX}\)[/tex] is [tex]\(\frac{3}{4}\)[/tex], and the original length of [tex]\(\overline{WX}\)[/tex] is 5. After dilation, the length is 15.
### Conclusion
The correct statement is:
C. The slope of [tex]\(\overline{W X}\)[/tex] is [tex]\(\frac{3}{4}\)[/tex], and the length of [tex]\(\overline{W X}\)[/tex] is 5.
