Question
line Ac is a dilation of AB, with point B as center of dilation. Determine the scale factor.
coordinate A(-1,-1)
coordinate B(2,2)
coordinate C(5,5)
coordinate A(-1,-1)
coordinate B(2,2)
coordinate C(5,5)
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Answered by
GPT-4o mini
To determine the scale factor of the dilation of line segment \( AB \) to line segment \( AC \) with point \( B \) as the center of dilation, we can use the formula for the scale factor, which is given by
\[
k = \frac{\text{distance from } B \text{ to } C}{\text{distance from } B \text{ to } A}
\]
First, we need to calculate the distances from point \( B \) to points \( A \) and \( C \).
**1. Calculate the distance from \( B \) to \( A \):**
Coordinates:
- \( A(-1, -1) \)
- \( B(2, 2) \)
Using the distance formula, we have:
\[
d_{BA} = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2}
\]
Substituting the coordinates:
\[
d_{BA} = \sqrt{(2 - (-1))^2 + (2 - (-1))^2} = \sqrt{(2 + 1)^2 + (2 + 1)^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\]
**2. Calculate the distance from \( B \) to \( C \):**
Coordinates:
- \( C(5, 5) \)
Again using the distance formula:
\[
d_{BC} = \sqrt{(x_B - x_C)^2 + (y_B - y_C)^2}
\]
Substituting the coordinates:
\[
d_{BC} = \sqrt{(2 - 5)^2 + (2 - 5)^2} = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\]
**3. Calculate the scale factor \( k \):**
Now substituting the distances into the scale factor formula:
\[
k = \frac{d_{BC}}{d_{BA}} = \frac{3\sqrt{2}}{3\sqrt{2}} = 1
\]
Thus, the scale factor of the dilation from \( AB \) to \( AC \) with \( B \) as the center of dilation is
\[
\boxed{1}
\]
\[
k = \frac{\text{distance from } B \text{ to } C}{\text{distance from } B \text{ to } A}
\]
First, we need to calculate the distances from point \( B \) to points \( A \) and \( C \).
**1. Calculate the distance from \( B \) to \( A \):**
Coordinates:
- \( A(-1, -1) \)
- \( B(2, 2) \)
Using the distance formula, we have:
\[
d_{BA} = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2}
\]
Substituting the coordinates:
\[
d_{BA} = \sqrt{(2 - (-1))^2 + (2 - (-1))^2} = \sqrt{(2 + 1)^2 + (2 + 1)^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\]
**2. Calculate the distance from \( B \) to \( C \):**
Coordinates:
- \( C(5, 5) \)
Again using the distance formula:
\[
d_{BC} = \sqrt{(x_B - x_C)^2 + (y_B - y_C)^2}
\]
Substituting the coordinates:
\[
d_{BC} = \sqrt{(2 - 5)^2 + (2 - 5)^2} = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\]
**3. Calculate the scale factor \( k \):**
Now substituting the distances into the scale factor formula:
\[
k = \frac{d_{BC}}{d_{BA}} = \frac{3\sqrt{2}}{3\sqrt{2}} = 1
\]
Thus, the scale factor of the dilation from \( AB \) to \( AC \) with \( B \) as the center of dilation is
\[
\boxed{1}
\]
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