Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Join our Q&A platform and connect with professionals ready to provide precise answers to your questions in various areas. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
To tackle the problem, we need to follow a series of steps involving the calculation of the new coordinates after dilation, the slope of the line, and the length of the segment.
### Step 1: Compute Coordinates of [tex]\(X'\)[/tex]
Given:
- [tex]\(W = (3, 2)\)[/tex]
- [tex]\(X = (7, 5)\)[/tex]
- Scale factor = 3
The coordinates of [tex]\(X'\)[/tex] after dilation with [tex]\(W\)[/tex] as the center of dilation are found using the formula:
[tex]\[ X' = W + \text{scale factor} \times (X - W) \][/tex]
#### Calculation:
- [tex]\(X'_{x}\)[/tex]: [tex]\(3 + 3 \times (7 - 3) = 3 + 3 \times 4 = 3 + 12 = 15\)[/tex]
- [tex]\(X'_{y}\)[/tex]: [tex]\(2 + 3 \times (5 - 2) = 2 + 3 \times 3 = 2 + 9 = 11\)[/tex]
Thus, the new coordinates of [tex]\(X'\)[/tex] are [tex]\((15, 11)\)[/tex].
### Step 2: Compute Slope of [tex]\(\overline{W'X'}\)[/tex]
The slope of a line through 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]
#### Calculation:
- For [tex]\(W = (3, 2)\)[/tex] and [tex]\(X' = (15, 11)\)[/tex]:
[tex]\[ \text{slope}_{W'X'} = \frac{11 - 2}{15 - 3} = \frac{9}{12} = \frac{3}{4} \][/tex]
### Step 3: Compute Length of [tex]\(\overline{W'X'}\)[/tex]
The length of a segment between points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ \text{length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
#### Calculation:
- For [tex]\(W = (3, 2)\)[/tex] and [tex]\(X' = (15, 11)\)[/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]
### Step 4: Select the Correct Statement
We compare our results to the options given:
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]\(\overline{WX'}\)[/tex] is [tex]\(\frac{3}{4}\)[/tex], and the length of [tex]\(\overline{WX'}\)[/tex] is 5 .
C. 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 .
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 .
Based on our calculations:
- 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]\( \boxed{D} \)[/tex].
### Step 1: Compute Coordinates of [tex]\(X'\)[/tex]
Given:
- [tex]\(W = (3, 2)\)[/tex]
- [tex]\(X = (7, 5)\)[/tex]
- Scale factor = 3
The coordinates of [tex]\(X'\)[/tex] after dilation with [tex]\(W\)[/tex] as the center of dilation are found using the formula:
[tex]\[ X' = W + \text{scale factor} \times (X - W) \][/tex]
#### Calculation:
- [tex]\(X'_{x}\)[/tex]: [tex]\(3 + 3 \times (7 - 3) = 3 + 3 \times 4 = 3 + 12 = 15\)[/tex]
- [tex]\(X'_{y}\)[/tex]: [tex]\(2 + 3 \times (5 - 2) = 2 + 3 \times 3 = 2 + 9 = 11\)[/tex]
Thus, the new coordinates of [tex]\(X'\)[/tex] are [tex]\((15, 11)\)[/tex].
### Step 2: Compute Slope of [tex]\(\overline{W'X'}\)[/tex]
The slope of a line through 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]
#### Calculation:
- For [tex]\(W = (3, 2)\)[/tex] and [tex]\(X' = (15, 11)\)[/tex]:
[tex]\[ \text{slope}_{W'X'} = \frac{11 - 2}{15 - 3} = \frac{9}{12} = \frac{3}{4} \][/tex]
### Step 3: Compute Length of [tex]\(\overline{W'X'}\)[/tex]
The length of a segment between points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ \text{length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
#### Calculation:
- For [tex]\(W = (3, 2)\)[/tex] and [tex]\(X' = (15, 11)\)[/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]
### Step 4: Select the Correct Statement
We compare our results to the options given:
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]\(\overline{WX'}\)[/tex] is [tex]\(\frac{3}{4}\)[/tex], and the length of [tex]\(\overline{WX'}\)[/tex] is 5 .
C. 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 .
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 .
Based on our calculations:
- 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]\( \boxed{D} \)[/tex].
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
