Dilations Keep Lines Parallel - Geometry
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What is the image of a line $\ell$ under a dilation with scale factor $k\neq 1$ if $\ell$ passes through $O$?
What is the image of a line $\ell$ under a dilation with scale factor $k\neq 1$ if $\ell$ passes through $O$?
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The same line $\ell$ (still unchanged). Lines through the center are always unchanged.
The same line $\ell$ (still unchanged). Lines through the center are always unchanged.
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State the line-mapping rule for dilation: lines through $O$ and lines not through $O$.
State the line-mapping rule for dilation: lines through $O$ and lines not through $O$.
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Through $O$: unchanged; not through $O$: mapped to a parallel line. This is the fundamental line-mapping property of dilation.
Through $O$: unchanged; not through $O$: mapped to a parallel line. This is the fundamental line-mapping property of dilation.
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What is true about angles formed by two lines after a dilation (assume both lines are not through $O$)?
What is true about angles formed by two lines after a dilation (assume both lines are not through $O$)?
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The angle measure is preserved. Parallel lines form angles with the same measure.
The angle measure is preserved. Parallel lines form angles with the same measure.
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What is the image of a segment lying on a line not through $O$ under a dilation with scale factor $k$?
What is the image of a segment lying on a line not through $O$ under a dilation with scale factor $k$?
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A segment on a line parallel to the original line. The original line maps to a parallel line.
A segment on a line parallel to the original line. The original line maps to a parallel line.
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What is the image of a segment lying on a line through $O$ under a dilation with scale factor $k$?
What is the image of a segment lying on a line through $O$ under a dilation with scale factor $k$?
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A segment on the same line through $O$. The segment lies on the unchanged line through $O$.
A segment on the same line through $O$. The segment lies on the unchanged line through $O$.
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Identify the correct statement: A dilation preserves line parallelism or makes all lines intersect?
Identify the correct statement: A dilation preserves line parallelism or makes all lines intersect?
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A dilation preserves parallelism of lines. Dilations preserve parallel relationships between lines.
A dilation preserves parallelism of lines. Dilations preserve parallel relationships between lines.
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What is the image of a vertical line $x=c$ under a dilation centered at $O=(0,0)$ with factor $k$?
What is the image of a vertical line $x=c$ under a dilation centered at $O=(0,0)$ with factor $k$?
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The line $x=kc$ (parallel to $x=c$). Vertical lines not through origin map to $x=kc$.
The line $x=kc$ (parallel to $x=c$). Vertical lines not through origin map to $x=kc$.
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Which condition guarantees that a dilation maps a line to itself: the line passes through $O$ or not?
Which condition guarantees that a dilation maps a line to itself: the line passes through $O$ or not?
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The line must pass through $O$. Only lines through the center are unchanged by dilation.
The line must pass through $O$. Only lines through the center are unchanged by dilation.
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What is the image of a line $\ell$ under a dilation if $\ell$ passes through the dilation center $O$?
What is the image of a line $\ell$ under a dilation if $\ell$ passes through the dilation center $O$?
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The same line $\ell$ (it is unchanged). Lines through the center remain fixed as sets.
The same line $\ell$ (it is unchanged). Lines through the center remain fixed as sets.
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What happens to the slope of a line under a dilation when the line passes through $O$?
What happens to the slope of a line under a dilation when the line passes through $O$?
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The slope is unchanged (the line is the same). The same line has the same slope.
The slope is unchanged (the line is the same). The same line has the same slope.
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What is the image of a line $\ell$ under a dilation if $\ell$ does not pass through the dilation center $O$?
What is the image of a line $\ell$ under a dilation if $\ell$ does not pass through the dilation center $O$?
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A line $\ell'$ parallel to $\ell$. Lines not through the center map to parallel lines.
A line $\ell'$ parallel to $\ell$. Lines not through the center map to parallel lines.
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What is the image of a horizontal line $y=c$ under a dilation centered at $O=(0,0)$ with factor $k$?
What is the image of a horizontal line $y=c$ under a dilation centered at $O=(0,0)$ with factor $k$?
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The line $y=kc$ (parallel to $y=c$). Horizontal lines not through origin map to $y=kc$.
The line $y=kc$ (parallel to $y=c$). Horizontal lines not through origin map to $y=kc$.
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What happens to the slope of a line under a dilation when the line does not pass through $O$?
What happens to the slope of a line under a dilation when the line does not pass through $O$?
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The slope is unchanged (the image line is parallel). Parallel lines have the same slope.
The slope is unchanged (the image line is parallel). Parallel lines have the same slope.
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What is the image of a line $\ell$ under a dilation with scale factor $k=1$?
What is the image of a line $\ell$ under a dilation with scale factor $k=1$?
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The same line $\ell$ (no change). Scale factor $k=1$ is the identity transformation.
The same line $\ell$ (no change). Scale factor $k=1$ is the identity transformation.
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Which condition guarantees that a dilation maps a line to a parallel line: the line passes through $O$ or not?
Which condition guarantees that a dilation maps a line to a parallel line: the line passes through $O$ or not?
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The line must not pass through $O$. Lines not through the center map to parallel lines.
The line must not pass through $O$. Lines not through the center map to parallel lines.
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Identify the invariant set for a dilation with center $O$: which lines are fixed pointwise as sets?
Identify the invariant set for a dilation with center $O$: which lines are fixed pointwise as sets?
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All lines that pass through $O$ are fixed as sets. Lines through the center are the only invariant sets.
All lines that pass through $O$ are fixed as sets. Lines through the center are the only invariant sets.
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What is true about the intersection point of two lines through $O$ after a dilation?
What is true about the intersection point of two lines through $O$ after a dilation?
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They still intersect at $O$ (the same point). Both lines pass through $O$ and remain unchanged.
They still intersect at $O$ (the same point). Both lines pass through $O$ and remain unchanged.
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Find the image of $x=-4$ under dilation centered at $O=(0,0)$ with scale factor $k=\frac{1}{2}$.
Find the image of $x=-4$ under dilation centered at $O=(0,0)$ with scale factor $k=\frac{1}{2}$.
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$x=-2$. Apply the formula: $x=kc$ where $k=\frac{1}{2}$ and $c=-4$.
$x=-2$. Apply the formula: $x=kc$ where $k=\frac{1}{2}$ and $c=-4$.
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Identify the image of $x=2$ under dilation centered at $O=(0,0)$ with scale factor $k=\frac{5}{2}$.
Identify the image of $x=2$ under dilation centered at $O=(0,0)$ with scale factor $k=\frac{5}{2}$.
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$x=5$. Apply the formula: $x = k \cdot 2 = \frac{5}{2} \cdot 2 = 5$.
$x=5$. Apply the formula: $x = k \cdot 2 = \frac{5}{2} \cdot 2 = 5$.
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Identify the image of $y=0x-4$ under dilation centered at $O=(0,0)$ with scale factor $k=3$.
Identify the image of $y=0x-4$ under dilation centered at $O=(0,0)$ with scale factor $k=3$.
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$y=-12$. The $y$-intercept becomes $k \cdot (-4) = 3 \cdot (-4) = -12$.
$y=-12$. The $y$-intercept becomes $k \cdot (-4) = 3 \cdot (-4) = -12$.
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Identify the image of $y=5x$ under dilation centered at $O=(0,0)$ with scale factor $k=\frac{7}{3}$.
Identify the image of $y=5x$ under dilation centered at $O=(0,0)$ with scale factor $k=\frac{7}{3}$.
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The same line $y=5x$. This line passes through the origin $(0,0)$.
The same line $y=5x$. This line passes through the origin $(0,0)$.
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Under dilation centered at $O=(0,0)$, what happens to the slope $m$ in $y=mx+b$?
Under dilation centered at $O=(0,0)$, what happens to the slope $m$ in $y=mx+b$?
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It stays $m$. Slope is preserved in the parallel image line.
It stays $m$. Slope is preserved in the parallel image line.
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Under dilation centered at $O=(0,0)$, what happens to the $y$-intercept $b$ in $y=mx+b$?
Under dilation centered at $O=(0,0)$, what happens to the $y$-intercept $b$ in $y=mx+b$?
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It becomes $kb$. The $y$-intercept is multiplied by the scale factor $k$.
It becomes $kb$. The $y$-intercept is multiplied by the scale factor $k$.
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A dilation centered at $O=(0,0)$ sends $y=mx+b$ to what line (assume $b\neq 0$)?
A dilation centered at $O=(0,0)$ sends $y=mx+b$ to what line (assume $b\neq 0$)?
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$y=mx+kb$. The slope stays $m$, the $y$-intercept becomes $kb$.
$y=mx+kb$. The slope stays $m$, the $y$-intercept becomes $kb$.
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Which option is correct for dilation: if $\ell \parallel m$, then $\ell' \parallel m'$?
Which option is correct for dilation: if $\ell \parallel m$, then $\ell' \parallel m'$?
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Yes, dilation preserves parallel lines: $\ell' \parallel m'$. Dilations map parallel lines to parallel lines.
Yes, dilation preserves parallel lines: $\ell' \parallel m'$. Dilations map parallel lines to parallel lines.
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Find the image of $y=-5$ under dilation centered at $O=(0,0)$ with scale factor $k=-2$.
Find the image of $y=-5$ under dilation centered at $O=(0,0)$ with scale factor $k=-2$.
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$y=10$. Apply the formula: $y=kc$ where $k=-2$ and $c=-5$.
$y=10$. Apply the formula: $y=kc$ where $k=-2$ and $c=-5$.
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Find the image of $x=7$ under dilation centered at $O=(0,0)$ with scale factor $k=-3$.
Find the image of $x=7$ under dilation centered at $O=(0,0)$ with scale factor $k=-3$.
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$x=-21$. Apply the formula: $x=kc$ where $k=-3$ and $c=7$.
$x=-21$. Apply the formula: $x=kc$ where $k=-3$ and $c=7$.
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A dilation is centered at $O=(0,0)$. Identify the image of the line $y=mx$ under any scale factor $k$.
A dilation is centered at $O=(0,0)$. Identify the image of the line $y=mx$ under any scale factor $k$.
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The same line $y=mx$. Lines through the origin are unchanged by dilation.
The same line $y=mx$. Lines through the origin are unchanged by dilation.
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A dilation is centered at $O=(0,0)$. Identify the image of the line $y=-3x$ under $k=5$.
A dilation is centered at $O=(0,0)$. Identify the image of the line $y=-3x$ under $k=5$.
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The same line $y=-3x$. This line passes through the origin $(0,0)$.
The same line $y=-3x$. This line passes through the origin $(0,0)$.
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A dilation is centered at $O=(0,0)$. Identify the image of the line $x=0$ under $k=-4$.
A dilation is centered at $O=(0,0)$. Identify the image of the line $x=0$ under $k=-4$.
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The same line $x=0$. The $y$-axis passes through the center of dilation.
The same line $x=0$. The $y$-axis passes through the center of dilation.
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