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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].
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