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Polygon WXYZ is dilated by a scale factor of 3 with vertex W as the center of dilation, resulting in polygon [tex]$W'X'Y'Z'$[/tex]. The coordinates of point W are [tex]$(3,2)$[/tex], and the coordinates of point [tex]$X$[/tex] are [tex]$(7,5)$[/tex].

Select the correct statement.

A. The slope of [tex]$\overline{WX}$[/tex] is [tex]$\frac{9}{4}$[/tex], and the length of [tex]$\overline{WX}$[/tex] is 15.
B. The slope of [tex]$\overline{WX}$[/tex] is [tex]$\frac{3}{4}$[/tex], and the length of [tex]$\overline{WX}$[/tex] is 15.
C. The slope of [tex]$\overline{WX}$[/tex] is [tex]$\frac{2}{4}$[/tex], and the length of [tex]$\overline{WX}$[/tex] is 5.
D. The slope of [tex]$\overline{WX}$[/tex] is [tex]$\frac{3}{4}$[/tex], and the length of [tex]$\overline{WX}$[/tex] is 5.


Sagot :

To solve this problem, we need to find the slope and length of the line segment [tex]\(\overline{WX}\)[/tex].

### Slope of [tex]\(\overline{WX}\)[/tex]

First, let's find the slope of [tex]\(\overline{WX}\)[/tex]. The formula to calculate the slope 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]

Here, the coordinates of point [tex]\(W\)[/tex] are [tex]\((3, 2)\)[/tex] and the coordinates of point [tex]\(X\)[/tex] are [tex]\((7, 5)\)[/tex].

Plugging the coordinates into the slope formula, we get:

[tex]\[ \text{slope} = \frac{5 - 2}{7 - 3} = \frac{3}{4} \][/tex]

### Length of [tex]\(\overline{WX}\)[/tex]

Next, we calculate the length of [tex]\(\overline{WX}\)[/tex] using the distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:

[tex]\[ \text{length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]

Substituting the coordinates of [tex]\(W\)[/tex] and [tex]\(X\)[/tex], we have:

[tex]\[ \text{length} = \sqrt{(7 - 3)^2 + (5 - 2)^2} \][/tex]

[tex]\[ \text{length} = \sqrt{4^2 + 3^2} \][/tex]

[tex]\[ \text{length} = \sqrt{16 + 9} \][/tex]

[tex]\[ \text{length} = \sqrt{25} \][/tex]

[tex]\[ \text{length} = 5 \][/tex]

With these calculations, we find that the slope of [tex]\(\overline{WX}\)[/tex] is [tex]\(\frac{3}{4}\)[/tex] and the length of [tex]\(\overline{WX}\)[/tex] is 5.

### Conclusion

So, the correct statement is:

D. 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.
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