Question
Which statement is part of the proof that the dilation of CD with a scale factor of 3
• and a center of dilation at
the origin will result in a line segment that is the length of CD?
(1 point)
•C" is at (9, -18) and D' is at (18, - 9).
• The length of C'D' is approximately 1.41 units.
OC" is at (2, -1) and D' is at (1, - 2).
• The length of C'D' is approximately 4.24.
C, 3,-6 D,6,-3
• and a center of dilation at
the origin will result in a line segment that is the length of CD?
(1 point)
•C" is at (9, -18) and D' is at (18, - 9).
• The length of C'D' is approximately 1.41 units.
OC" is at (2, -1) and D' is at (1, - 2).
• The length of C'D' is approximately 4.24.
C, 3,-6 D,6,-3
Answers
Answer
Whats the answer
Answered by
GPT-4o mini
To determine which statement is part of the proof regarding the dilation of the line segment CD with a scale factor of 3 and a center of dilation at the origin, we need to establish some preliminary facts about dilations and the points given.
Given the points C(3, -6) and D(6, -3), we can first calculate the length of the segment CD.
The length \( L \) of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the distance formula:
\[
L = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Plugging in the coordinates of C and D:
\[
L = \sqrt{(6 - 3)^2 + (-3 - (-6))^2} = \sqrt{(3)^2 + (3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \approx 4.24
\]
Next, let's analyze the statements provided:
1. \( C' \) is at (9, -18) and \( D' \) is at (18, -9).
2. The length of \( C'D' \) is approximately 1.41 units.
3. \( C' \) is at (2, -1) and \( D' \) is at (1, -2).
4. The length of \( C'D' \) is approximately 4.24.
Since we dilated the segment with a scale factor of 3, the length of the new line segment \( C'D' \) should be 3 times the original length \( L \).
Therefore, since the original length was approximately 4.24 units, the length after dilation is:
\[
3 \times 4.24 \approx 12.72 \text{ units }
\]
Thus, we need to verify which statement correctly describes the dilation. The statement that mentions the length of \( C'D' \) being approximately 4.24 reflects the length of the original line segment before dilation.
The correct statements in the context of the dilation (which should discuss the scaled length resulting from a dilation by a factor of 3) should not include lengths less than the original post-dilation factor. Thus, none of the statements explicitly represent correctly positioned dilation outputs based on a center at the origin.
However, based on the provided statements, **Statement 4** correctly notes the length of the dilated segment \( C'D' \) as approximately 4.24, matching our calculations. Therefore, the relevant statement should be:
- The length of \( C'D' \) is approximately 4.24.
So, the correct answer aligns with your description of lengths after the dilation; it's critical to note that there is inconsistency in the prior context established. If we assess that \( C' = (9, -18) \) and \( D' = (18, -9) \) are the results of applying the scale correctly, their length should at least scale correctly to \( \approx 12.72 \) units through direct distance comparisons, yet this outlines real checks with distance confirmation after the transformations.
Given the points C(3, -6) and D(6, -3), we can first calculate the length of the segment CD.
The length \( L \) of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the distance formula:
\[
L = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Plugging in the coordinates of C and D:
\[
L = \sqrt{(6 - 3)^2 + (-3 - (-6))^2} = \sqrt{(3)^2 + (3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \approx 4.24
\]
Next, let's analyze the statements provided:
1. \( C' \) is at (9, -18) and \( D' \) is at (18, -9).
2. The length of \( C'D' \) is approximately 1.41 units.
3. \( C' \) is at (2, -1) and \( D' \) is at (1, -2).
4. The length of \( C'D' \) is approximately 4.24.
Since we dilated the segment with a scale factor of 3, the length of the new line segment \( C'D' \) should be 3 times the original length \( L \).
Therefore, since the original length was approximately 4.24 units, the length after dilation is:
\[
3 \times 4.24 \approx 12.72 \text{ units }
\]
Thus, we need to verify which statement correctly describes the dilation. The statement that mentions the length of \( C'D' \) being approximately 4.24 reflects the length of the original line segment before dilation.
The correct statements in the context of the dilation (which should discuss the scaled length resulting from a dilation by a factor of 3) should not include lengths less than the original post-dilation factor. Thus, none of the statements explicitly represent correctly positioned dilation outputs based on a center at the origin.
However, based on the provided statements, **Statement 4** correctly notes the length of the dilated segment \( C'D' \) as approximately 4.24, matching our calculations. Therefore, the relevant statement should be:
- The length of \( C'D' \) is approximately 4.24.
So, the correct answer aligns with your description of lengths after the dilation; it's critical to note that there is inconsistency in the prior context established. If we assess that \( C' = (9, -18) \) and \( D' = (18, -9) \) are the results of applying the scale correctly, their length should at least scale correctly to \( \approx 12.72 \) units through direct distance comparisons, yet this outlines real checks with distance confirmation after the transformations.
Answered by
GPT-4o mini
The correct statement that is part of the proof that the dilation of CD with a scale factor of 3 results in a line segment from point C to point D (CD) with length approximately equal to the original segment CD (when considering the scale) is:
**The length of \( C'D' \) is approximately 4.24.**
However, note that this statement does not reflect the correct outcome after dilation. The length after a dilation by a factor of 3 should be about 12.72 units based on the original segment calculated, but among the options provided, that was the only relevant one mentioned regarding length.
If pressed to pick among choices purely based on format presented, that would ideally reference the length comparison you'd check against outcomes expected through dilation. If your aim is purely based on academic completeness, emphasizing the assertion of factor completion post was the ultimate target to reflect totality reporting shifting.
**The length of \( C'D' \) is approximately 4.24.**
However, note that this statement does not reflect the correct outcome after dilation. The length after a dilation by a factor of 3 should be about 12.72 units based on the original segment calculated, but among the options provided, that was the only relevant one mentioned regarding length.
If pressed to pick among choices purely based on format presented, that would ideally reference the length comparison you'd check against outcomes expected through dilation. If your aim is purely based on academic completeness, emphasizing the assertion of factor completion post was the ultimate target to reflect totality reporting shifting.
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