To solve this problem, let’s walk through the process of applying a dilation to the point [tex]\( X \)[/tex] and then calculating the relevant properties:
1. Initial Coordinates and Scale Factor:
- The initial coordinates of point [tex]\( X \)[/tex] are [tex]\( (7, 5) \)[/tex].
- The scale factor for the dilation is [tex]\( 3 \)[/tex].
2. Calculating New Coordinates After Dilation:
Since [tex]\( W \)[/tex] is the center of dilation and it's the origin [tex]\( (0,0) \)[/tex], the coordinates of [tex]\( X \)[/tex] after dilation (let's call it [tex]\( X' \)[/tex]) can be calculated by multiplying each coordinate of [tex]\( X \)[/tex] by the scale factor.
- New [tex]\( x \)[/tex]-coordinate: [tex]\( 7 \times 3 = 21 \)[/tex]
- New [tex]\( y \)[/tex]-coordinate: [tex]\( 5 \times 3 = 15 \)[/tex]
So, the new coordinates [tex]\( X' \)[/tex] are [tex]\( (21, 15) \)[/tex].
3. Calculating the Slope of [tex]\( \overline{WX} \)[/tex] and [tex]\( \overline{WX'} \)[/tex]:
The slope of a line from point [tex]\( W \)[/tex] (the origin) to point [tex]\( X \)[/tex] is calculated by dividing the [tex]\( y \)[/tex]-coordinate by the [tex]\( x \)[/tex]-coordinate.
- Initial slope: [tex]\( \frac{5}{7} \)[/tex]
- New slope after dilation (since dilation does not change the ratio): [tex]\( \frac{15}{21} = \frac{5}{7} \)[/tex] which is approximately 0.7142857142857143 in decimal form.
4. Calculating the Length of the Segment [tex]\( \overline{WX} \)[/tex] and [tex]\( \overline{WX'} \)[/tex]:
We use the distance formula to find the length of the segment.
[tex]\[
\text{Initial length of } \overline{WX} = \sqrt{7^2 + 5^2} = \sqrt{49 + 25} = \sqrt{74} \approx 8.602325267042627
\][/tex]
- After dilation, the new length: [tex]\( 8.602325267042627 \times 3 \approx 25.80697580112788 \)[/tex]
5. Comparing with the Given Statements:
- The slope of [tex]\( \overline{W'X'} \)[/tex] is given approximately as 0.7142857142857143, which corresponds to [tex]\(\frac{5}{7}\)[/tex] rather than [tex]\(\frac{3}{4}\)[/tex] or [tex]\(\frac{9}{4}\)[/tex].
- The length of [tex]\( \overline{W'X'} \)[/tex] is approximately 25.807.
Let’s review the given statements:
- 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 5. (Incorrect)
- 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 15. (Incorrect)
- 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. (Incorrect)
- 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 15. (Incorrect)
None of the given statements correctly describe both the slope and the length of the dilated segment. Therefore, the correct conclusion is that none of the provided options are correct.
