Polygon WXYZ is dilated by a scale factor of 3 with vertex [tex]$W$[/tex] as the center of dilation, resulting in polygon [tex]$W'X'Y'Z'$[/tex]. 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].

Select the correct statement.

A. The slope of [tex]$\overline{W'X'}$[/tex] is [tex]$\frac{9}{4}$[/tex], and the length of [tex]$\overline{W'X'}$[/tex] is 15.

B. The slope of [tex]$\overline{W'X'}$[/tex] is [tex]$\frac{9}{4}$[/tex], and the length of [tex]$\overline{W'X'}$[/tex] is 5.

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 15.

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.



Answer :

To solve the problem, follow these steps:

1. Identify the coordinates of points W and X:
[tex]\[ W = (3, 2) \quad \text{and} \quad X = (7, 5) \][/tex]

2. Calculate the slope of line segment [tex]\( \overline{WX} \)[/tex]:
The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ \text{slope}_{WX} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the values:
[tex]\[ \text{slope}_{WX} = \frac{5 - 2}{7 - 3} = \frac{3}{4} \][/tex]

3. Dilate point X with respect to W with a scale factor of 3:
The formula to find the coordinates of the dilated point [tex]\(X'\)[/tex] is given by:
[tex]\[ X' = \left(W_x + \text{scale factor} \times (X_x - W_x), W_y + \text{scale factor} \times (X_y - W_y)\right) \][/tex]
Plugging in the values:
[tex]\[ X' = \left(3 + 3 \times (7 - 3), 2 + 3 \times (5 - 2)\right) = (3 + 12, 2 + 9) = (15, 11) \][/tex]

4. Calculate the slope of line segment [tex]\( \overline{W'X'} \)[/tex]:
Since W' coincides with W (because they are the center of dilation) the slope of [tex]\( \overline{W'X'} \)[/tex] is the same as the slope of [tex]\( \overline{WX} \)[/tex]:
[tex]\[ \text{slope}_{W'X'} = \frac{11 - 2}{15 - 3} = \frac{9}{12} = \frac{3}{4} \][/tex]
So, the slope of [tex]\( \overline{W'X'} \)[/tex] remains:
[tex]\[ \text{slope}_{W'X'} = \frac{3}{4} \][/tex]

5. Calculate the length of [tex]\( \overline{W'X'} \)[/tex]:
Using the distance formula to calculate the length of [tex]\( \overline{W'X'} \)[/tex]:
[tex]\[ \text{length}_{W'X'} = \sqrt{(15 - 3)^2 + (11 - 2)^2} = \sqrt{12^2 + 9^2} = \sqrt{144 + 81} = \sqrt{225} = 15 \][/tex]

6. Identify the correct statement:
We need to match the slope of [tex]\( \overline{W'X'} \)[/tex] and the length of [tex]\( \overline{W'X'} \)[/tex] with the given options.

- We found that the slope of [tex]\( \overline{W'X'} \)[/tex] is [tex]\(\frac{3}{4}\)[/tex].
- The length of [tex]\( \overline{W'X'} \)[/tex] is 15.

Thus, the correct statement is:
[tex]\[ \boxed{\text{C. The slope of \(\overline{W'X'}\) is \(\frac{3}{4}\), and the length of \(\overline{W'X'}\) is 15.}} \][/tex]