Understand Parallel Line Transformation Properties - 8th Grade Math
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What must be true about the images of two parallel lines after a translation?
What must be true about the images of two parallel lines after a translation?
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If $l_1 \parallel l_2$, then $T(l_1) \parallel T(l_2)$. Translations preserve slopes, so parallel lines remain parallel.
If $l_1 \parallel l_2$, then $T(l_1) \parallel T(l_2)$. Translations preserve slopes, so parallel lines remain parallel.
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Decide if the lines stay parallel: $l: y=-3x+4$ and $m: y=-3x-2$ rotated $90^\circ$ about the origin.
Decide if the lines stay parallel: $l: y=-3x+4$ and $m: y=-3x-2$ rotated $90^\circ$ about the origin.
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Yes, the images are parallel. Rotation preserves parallelism regardless of slope.
Yes, the images are parallel. Rotation preserves parallelism regardless of slope.
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Identify the error: A student claims a translation can make parallel lines intersect. What is the correction?
Identify the error: A student claims a translation can make parallel lines intersect. What is the correction?
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A translation cannot make parallel lines intersect. Rigid motions preserve parallelism always.
A translation cannot make parallel lines intersect. Rigid motions preserve parallelism always.
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What does it mean for two lines $l$ and $m$ to be parallel?
What does it mean for two lines $l$ and $m$ to be parallel?
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$l \parallel m$ means they never intersect. Parallel lines maintain constant distance apart.
$l \parallel m$ means they never intersect. Parallel lines maintain constant distance apart.
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Which rigid motions are included in CCSS.8.G.1.c for mapping parallel lines to parallel lines?
Which rigid motions are included in CCSS.8.G.1.c for mapping parallel lines to parallel lines?
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Translations, reflections, and rotations. These three rigid motions preserve parallelism.
Translations, reflections, and rotations. These three rigid motions preserve parallelism.
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Find the image slope: If a line has slope $m$, what is the slope after a $180^\circ$ rotation about the origin?
Find the image slope: If a line has slope $m$, what is the slope after a $180^\circ$ rotation about the origin?
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$m$. $180°$ rotation preserves slope magnitude and sign.
$m$. $180°$ rotation preserves slope magnitude and sign.
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Find the image slope: If a line has slope $m$, what is the slope after reflection across the $y$-axis?
Find the image slope: If a line has slope $m$, what is the slope after reflection across the $y$-axis?
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$-m$. Reflecting across $y$-axis negates the slope.
$-m$. Reflecting across $y$-axis negates the slope.
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Find the image slope: If a line has slope $m$, what is the slope after reflection across the $x$-axis?
Find the image slope: If a line has slope $m$, what is the slope after reflection across the $x$-axis?
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$-m$. Reflecting across $x$-axis negates the slope.
$-m$. Reflecting across $x$-axis negates the slope.
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Identify the transformation type: $(x,y)\rightarrow(-x,y)$.
Identify the transformation type: $(x,y)\rightarrow(-x,y)$.
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Reflection across the $y$-axis. Flips points across the vertical axis.
Reflection across the $y$-axis. Flips points across the vertical axis.
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Identify the transformation type: $(x,y)\rightarrow(y,-x)$.
Identify the transformation type: $(x,y)\rightarrow(y,-x)$.
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Rotation $90^\circ$ clockwise about the origin. Rotates points $90°$ clockwise around $(0,0)$.
Rotation $90^\circ$ clockwise about the origin. Rotates points $90°$ clockwise around $(0,0)$.
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Find the image slope: If a line has slope $m$, what is the slope after translation $(x,y)\rightarrow(x+3,y-2)$?
Find the image slope: If a line has slope $m$, what is the slope after translation $(x,y)\rightarrow(x+3,y-2)$?
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$m$. Translations don't change slope direction.
$m$. Translations don't change slope direction.
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What is the key conclusion of CCSS.8.G.1.c about images of parallel lines under rigid motions?
What is the key conclusion of CCSS.8.G.1.c about images of parallel lines under rigid motions?
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Parallel lines map to parallel lines. Rigid motions preserve the parallel relationship.
Parallel lines map to parallel lines. Rigid motions preserve the parallel relationship.
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Decide if the lines stay parallel: $l: y=2x+1$ and $m: y=2x-5$ reflected across the $x$-axis.
Decide if the lines stay parallel: $l: y=2x+1$ and $m: y=2x-5$ reflected across the $x$-axis.
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Yes, the images are parallel. Both lines have slope $2$, so images have slope $-2$.
Yes, the images are parallel. Both lines have slope $2$, so images have slope $-2$.
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What must be true about the images of two parallel lines after any rigid motion in the plane?
What must be true about the images of two parallel lines after any rigid motion in the plane?
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They remain parallel and do not intersect. Rigid motions preserve the parallel relationship.
They remain parallel and do not intersect. Rigid motions preserve the parallel relationship.
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Identify the transformation type: $(x,y)\rightarrow(x+a,y+b)$.
Identify the transformation type: $(x,y)\rightarrow(x+a,y+b)$.
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Translation. Shifts all points by constant amounts $a$ and $b$.
Translation. Shifts all points by constant amounts $a$ and $b$.
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What is the slope of the image of $y = 4x - 9$ after a reflection across the $x$-axis?
What is the slope of the image of $y = 4x - 9$ after a reflection across the $x$-axis?
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$-4$. Reflection across $x$-axis negates the slope.
$-4$. Reflection across $x$-axis negates the slope.
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What is the slope of the image of $y = 2x + 1$ after a translation by $(5, -3)$?
What is the slope of the image of $y = 2x + 1$ after a translation by $(5, -3)$?
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$2$. Translations don't change slope; shifts position only.
$2$. Translations don't change slope; shifts position only.
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What is the slope of the image of $y = -2x + 1$ after a $180^\circ$ rotation about the origin?
What is the slope of the image of $y = -2x + 1$ after a $180^\circ$ rotation about the origin?
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$-2$. $180°$ rotation preserves slope magnitude and sign.
$-2$. $180°$ rotation preserves slope magnitude and sign.
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Identify the correct conclusion: If two lines are not parallel, can a rigid motion make them parallel?
Identify the correct conclusion: If two lines are not parallel, can a rigid motion make them parallel?
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No; rigid motions preserve angles, so nonparallel stays nonparallel. Rigid motions preserve all geometric relationships.
No; rigid motions preserve angles, so nonparallel stays nonparallel. Rigid motions preserve all geometric relationships.
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What is the image relationship: If $l_1 \perp l_2$, what is true about $M(l_1)$ and $M(l_2)$ for a rigid motion $M$?
What is the image relationship: If $l_1 \perp l_2$, what is true about $M(l_1)$ and $M(l_2)$ for a rigid motion $M$?
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They stay perpendicular: $M(l_1) \perp M(l_2)$. Rigid motions preserve perpendicularity ($90°$ angles).
They stay perpendicular: $M(l_1) \perp M(l_2)$. Rigid motions preserve perpendicularity ($90°$ angles).
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Choose the word that completes the statement: Rigid motions preserve distances and ____.
Choose the word that completes the statement: Rigid motions preserve distances and ____.
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angle measures. Rigid motions are isometries preserving all measurements.
angle measures. Rigid motions are isometries preserving all measurements.
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What is the image relationship: If $l_1 \parallel l_2$, what is true about the distance between them after a rigid motion?
What is the image relationship: If $l_1 \parallel l_2$, what is true about the distance between them after a rigid motion?
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The distance between the lines stays the same. Rigid motions are distance-preserving transformations.
The distance between the lines stays the same. Rigid motions are distance-preserving transformations.
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Find the correct statement: A reflection can turn two parallel lines into the same line. True or false?
Find the correct statement: A reflection can turn two parallel lines into the same line. True or false?
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False; distinct parallel lines remain distinct parallel lines. Rigid motions preserve distinctness of geometric objects.
False; distinct parallel lines remain distinct parallel lines. Rigid motions preserve distinctness of geometric objects.
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Identify whether the lines stay parallel: If $l_1 \parallel l_2$, are $R_{45^\circ}(l_1)$ and $R_{45^\circ}(l_2)$ parallel?
Identify whether the lines stay parallel: If $l_1 \parallel l_2$, are $R_{45^\circ}(l_1)$ and $R_{45^\circ}(l_2)$ parallel?
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Yes, $R_{45^\circ}(l_1) \parallel R_{45^\circ}(l_2)$. Any rotation preserves parallelism between lines.
Yes, $R_{45^\circ}(l_1) \parallel R_{45^\circ}(l_2)$. Any rotation preserves parallelism between lines.
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What is the rigid motion property about parallel lines under translations, rotations, and reflections?
What is the rigid motion property about parallel lines under translations, rotations, and reflections?
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Parallel lines map to parallel lines under any rigid motion. Rigid motions preserve distances and angles, maintaining parallel relationships.
Parallel lines map to parallel lines under any rigid motion. Rigid motions preserve distances and angles, maintaining parallel relationships.
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