Question
Which statement is part of the proof that the dilation of CD¯¯¯¯¯¯¯¯
with a scale factor of 13
and a center of dilation at the origin will result in a line segment that is 13
the length of CD¯¯¯¯¯¯¯¯
?
(1 point)
Responses
C′
is at (2,−1)
and D′
is at (1,−2)
.
upper C prime is at open paren 2 comma negative 1 close paren and upper D prime is at left parenthesis 1 comma negative 2 right parenthesis .
The length of C′D′¯¯¯¯¯¯¯¯¯¯¯
is approximately 4.24.
The length of line segment cap c prime cap d prime is approximately 4.24.
The length of C′D′¯¯¯¯¯¯¯¯¯¯¯
is approximately 1.41 units.
The length of line segment cap c prime cap d prime is approximately 1.41 units.
C′
is at (9,−18)
and D′
is at (18,−9)
.
upper C prime is at open paren 9 comma negative 18 close paren and upper D prime is at open paren 18 comma negative 9 close paren .
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with a scale factor of 13
and a center of dilation at the origin will result in a line segment that is 13
the length of CD¯¯¯¯¯¯¯¯
?
(1 point)
Responses
C′
is at (2,−1)
and D′
is at (1,−2)
.
upper C prime is at open paren 2 comma negative 1 close paren and upper D prime is at left parenthesis 1 comma negative 2 right parenthesis .
The length of C′D′¯¯¯¯¯¯¯¯¯¯¯
is approximately 4.24.
The length of line segment cap c prime cap d prime is approximately 4.24.
The length of C′D′¯¯¯¯¯¯¯¯¯¯¯
is approximately 1.41 units.
The length of line segment cap c prime cap d prime is approximately 1.41 units.
C′
is at (9,−18)
and D′
is at (18,−9)
.
upper C prime is at open paren 9 comma negative 18 close paren and upper D prime is at open paren 18 comma negative 9 close paren .
Skip to navigation
Answers
GPT-4o mini
To prove that the dilation of line segment \( \overline{CD} \) with a scale factor of 13 and a center of dilation at the origin results in a line segment that is 13 times the length of \( \overline{CD} \), we must understand how dilation affects the length of line segments.
Dilation transforms a point \((x, y)\) at the origin with a scale factor \(k\) to the point \((kx, ky)\). Consequently, for a segment \( \overline{CD} \), if we denote the endpoints of \(CD\) as \(C(x_1, y_1)\) and \(D(x_2, y_2)\), after the dilation with a scale factor of 13, the new endpoints become:
\[
C' = (13x_1, 13y_1)
\]
\[
D' = (13x_2, 13y_2)
\]
Now, the length of \( \overline{CD} \) can be calculated using the distance formula:
\[
\text{Length of } \overline{CD} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
After dilation, the length of \( \overline{C'D'} \) becomes:
\[
\text{Length of } \overline{C'D'} = \sqrt{(13x_2 - 13x_1)^2 + (13y_2 - 13y_1)^2} = 13\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
This demonstrates that the length of \( \overline{C'D'}\) is indeed 13 times the length of \( \overline{CD}\).
From the options provided, we look for a statement that aids in establishing that \( \overline{C'D'}\) is effectively 13 times longer than \( \overline{CD}\).
The statement confirming the transformed endpoints, \(C'\) and \(D'\), provides essential support for proof. Specifically, knowing the locations of \(C'\) and \(D'\) allows us to assess the length post-dilation directly.
Thus, the correct response that contributes to the proof is:
**C′ is at (9,−18) and D′ is at (18,−9).**
This statement describes the new locations of the endpoints after dilation with a scale factor of 13, which is key to confirming that the resulting line segment's length is indeed proportional to the original length by the said factor.
Dilation transforms a point \((x, y)\) at the origin with a scale factor \(k\) to the point \((kx, ky)\). Consequently, for a segment \( \overline{CD} \), if we denote the endpoints of \(CD\) as \(C(x_1, y_1)\) and \(D(x_2, y_2)\), after the dilation with a scale factor of 13, the new endpoints become:
\[
C' = (13x_1, 13y_1)
\]
\[
D' = (13x_2, 13y_2)
\]
Now, the length of \( \overline{CD} \) can be calculated using the distance formula:
\[
\text{Length of } \overline{CD} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
After dilation, the length of \( \overline{C'D'} \) becomes:
\[
\text{Length of } \overline{C'D'} = \sqrt{(13x_2 - 13x_1)^2 + (13y_2 - 13y_1)^2} = 13\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
This demonstrates that the length of \( \overline{C'D'}\) is indeed 13 times the length of \( \overline{CD}\).
From the options provided, we look for a statement that aids in establishing that \( \overline{C'D'}\) is effectively 13 times longer than \( \overline{CD}\).
The statement confirming the transformed endpoints, \(C'\) and \(D'\), provides essential support for proof. Specifically, knowing the locations of \(C'\) and \(D'\) allows us to assess the length post-dilation directly.
Thus, the correct response that contributes to the proof is:
**C′ is at (9,−18) and D′ is at (18,−9).**
This statement describes the new locations of the endpoints after dilation with a scale factor of 13, which is key to confirming that the resulting line segment's length is indeed proportional to the original length by the said factor.