Question
Is A′B′¯¯¯¯¯¯¯¯¯¯ a dilation of AB¯¯¯¯¯¯¯¯ ? If yes, what is the scale factor and is the dilation an enlargement or a reduction? (1 point) Responses No, A′B′¯¯¯¯¯¯¯¯¯¯ is not a dilation of AB¯¯¯¯¯¯¯¯ No, line segment cap A prime cap b prime is not a dilation of line segment cap A cap b Yes, A′B′¯¯¯¯¯¯¯¯¯¯ is a dilation of AB¯¯¯¯¯¯¯¯ , with a scale factor of 2, which is an enlargement. Yes, line segment cap A prime cap b prime is a dilation of line segment cap A cap b , with a scale factor of 2, which is an enlargement. Yes, A′B′¯¯¯¯¯¯¯¯¯¯ is a dilation of AB¯¯¯¯¯¯¯¯ , with a scale factor of 12 , which is an enlargement. Yes, line segment cap A prime cap b prime is a dilation of line segment cap A cap b , with a scale factor of 1 half, which is an enlargement. Yes, A′B′¯¯¯¯¯¯¯¯¯¯ is a dilation of AB¯¯¯¯¯¯¯¯ , with a scale factor of 12 , which is a reduction. A -2, 0.5 B 0, -0.5 A' -4, 1 B' 0, -1
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Answered by
GPT-4o mini
To determine if \( A'B' \) is a dilation of \( AB \), we first need to calculate the lengths of the segments \( AB \) and \( A'B' \), and then check for the scale factor.
**Step 1: Finding the Length of Segment \( AB \)**
Given points:
- \( A (-2, 0.5) \)
- \( B (0, -0.5) \)
The length of segment \( AB \) can be calculated using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting the coordinates:
\[
d_{AB} = \sqrt{(0 - (-2))^2 + (-0.5 - 0.5)^2}
= \sqrt{(2)^2 + (-1)^2}
= \sqrt{4 + 1}
= \sqrt{5}
\]
**Step 2: Finding the Length of Segment \( A'B' \)**
Given points:
- \( A' (-4, 1) \)
- \( B' (0, -1) \)
Calculating the length of segment \( A'B' \):
\[
d_{A'B'} = \sqrt{(0 - (-4))^2 + (-1 - 1)^2}
= \sqrt{(4)^2 + (-2)^2}
= \sqrt{16 + 4}
= \sqrt{20}
= 2\sqrt{5}
\]
**Step 3: Determine the Scale Factor**
To find the scale factor \( k \), we compare the lengths of \( A'B' \) and \( AB \):
\[
k = \frac{d_{A'B'}}{d_{AB}} = \frac{2\sqrt{5}}{\sqrt{5}} = 2
\]
**Step 4: Determine if it is an Enlargement or a Reduction**
Since the scale factor \( k = 2 \) is greater than 1, this indicates that \( A'B' \) is an enlargement of \( AB \).
**Conclusion**
Yes, \( A'B' \) is a dilation of \( AB \), with a scale factor of 2, which is an enlargement.
Therefore, the correct response is:
**Yes, A′B′ is a dilation of AB, with a scale factor of 2, which is an enlargement.**
**Step 1: Finding the Length of Segment \( AB \)**
Given points:
- \( A (-2, 0.5) \)
- \( B (0, -0.5) \)
The length of segment \( AB \) can be calculated using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting the coordinates:
\[
d_{AB} = \sqrt{(0 - (-2))^2 + (-0.5 - 0.5)^2}
= \sqrt{(2)^2 + (-1)^2}
= \sqrt{4 + 1}
= \sqrt{5}
\]
**Step 2: Finding the Length of Segment \( A'B' \)**
Given points:
- \( A' (-4, 1) \)
- \( B' (0, -1) \)
Calculating the length of segment \( A'B' \):
\[
d_{A'B'} = \sqrt{(0 - (-4))^2 + (-1 - 1)^2}
= \sqrt{(4)^2 + (-2)^2}
= \sqrt{16 + 4}
= \sqrt{20}
= 2\sqrt{5}
\]
**Step 3: Determine the Scale Factor**
To find the scale factor \( k \), we compare the lengths of \( A'B' \) and \( AB \):
\[
k = \frac{d_{A'B'}}{d_{AB}} = \frac{2\sqrt{5}}{\sqrt{5}} = 2
\]
**Step 4: Determine if it is an Enlargement or a Reduction**
Since the scale factor \( k = 2 \) is greater than 1, this indicates that \( A'B' \) is an enlargement of \( AB \).
**Conclusion**
Yes, \( A'B' \) is a dilation of \( AB \), with a scale factor of 2, which is an enlargement.
Therefore, the correct response is:
**Yes, A′B′ is a dilation of AB, with a scale factor of 2, which is an enlargement.**
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