Understand Congruence Through Transformations - 8th Grade Math
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What does it mean for two plane figures to be congruent using rigid motions?
What does it mean for two plane figures to be congruent using rigid motions?
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One can be obtained from the other by translations, rotations, and reflections. Rigid motions preserve size and shape while moving figures.
One can be obtained from the other by translations, rotations, and reflections. Rigid motions preserve size and shape while moving figures.
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What is the image of $P(3,-4)$ after reflection across the $y$-axis?
What is the image of $P(3,-4)$ after reflection across the $y$-axis?
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$P'(-3,-4)$. Reflect across $y$-axis: negate $x$-coordinate.
$P'(-3,-4)$. Reflect across $y$-axis: negate $x$-coordinate.
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What is the image of $P(-2,5)$ after the translation $(x,y)\rightarrow(x+4,y-3)$?
What is the image of $P(-2,5)$ after the translation $(x,y)\rightarrow(x+4,y-3)$?
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$P'(2,2)$. $(-2+4, 5-3) = (2,2)$.
$P'(2,2)$. $(-2+4, 5-3) = (2,2)$.
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Identify the transformation: $(x,y)\rightarrow(x,-y)$.
Identify the transformation: $(x,y)\rightarrow(x,-y)$.
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Reflection across the $x$-axis. Negating $y$ flips points across the $x$-axis.
Reflection across the $x$-axis. Negating $y$ flips points across the $x$-axis.
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Identify the transformation: $(x,y)\rightarrow(-x,y)$.
Identify the transformation: $(x,y)\rightarrow(-x,y)$.
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Reflection across the $y$-axis. Negating $x$ flips points across the $y$-axis.
Reflection across the $y$-axis. Negating $x$ flips points across the $y$-axis.
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Identify the transformation: $(x,y)\rightarrow(y,-x)$.
Identify the transformation: $(x,y)\rightarrow(y,-x)$.
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Rotation $90^\circ$ clockwise about the origin. $(x,y) o (y,-x)$ rotates $90°$ clockwise.
Rotation $90^\circ$ clockwise about the origin. $(x,y) o (y,-x)$ rotates $90°$ clockwise.
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Identify the transformation: $(x,y)\rightarrow(-x,-y)$.
Identify the transformation: $(x,y)\rightarrow(-x,-y)$.
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Rotation $180^\circ$ about the origin. Negating both coordinates rotates $180°$.
Rotation $180^\circ$ about the origin. Negating both coordinates rotates $180°$.
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Identify the transformation: $(x,y)\rightarrow(x+3,y-2)$.
Identify the transformation: $(x,y)\rightarrow(x+3,y-2)$.
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Translation right $3$ and down $2$. Add $3$ to $x$, subtract $2$ from $y$.
Translation right $3$ and down $2$. Add $3$ to $x$, subtract $2$ from $y$.
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What is a rotation of a figure in the plane?
What is a rotation of a figure in the plane?
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A turn about a fixed center by a specified angle and direction. Points rotate around a center point.
A turn about a fixed center by a specified angle and direction. Points rotate around a center point.
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What is the image of $P(1,6)$ after a $90^\circ$ counterclockwise rotation about the origin?
What is the image of $P(1,6)$ after a $90^\circ$ counterclockwise rotation about the origin?
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$P'(-6,1)$. $90°$ counterclockwise: $(1,6) o (-6,1)$.
$P'(-6,1)$. $90°$ counterclockwise: $(1,6) o (-6,1)$.
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What is a translation of a figure in the plane?
What is a translation of a figure in the plane?
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A slide that moves every point the same distance in the same direction. Every point moves by the same vector.
A slide that moves every point the same distance in the same direction. Every point moves by the same vector.
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Which sequence maps a figure to its mirror image across the $y$-axis without changing size?
Which sequence maps a figure to its mirror image across the $y$-axis without changing size?
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Reflect the figure across the $y$-axis. Reflection preserves size while creating mirror image.
Reflect the figure across the $y$-axis. Reflection preserves size while creating mirror image.
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What is the minimum information needed to describe a rotation precisely?
What is the minimum information needed to describe a rotation precisely?
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Center of rotation, angle measure, and direction. Must specify where, how much, and which way to rotate.
Center of rotation, angle measure, and direction. Must specify where, how much, and which way to rotate.
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Which transformations are rigid motions that preserve distance and angle measure?
Which transformations are rigid motions that preserve distance and angle measure?
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Translations, rotations, and reflections. These transformations preserve distances and angles.
Translations, rotations, and reflections. These transformations preserve distances and angles.
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Identify the transformation: $(x,y)\rightarrow(-y,x)$.
Identify the transformation: $(x,y)\rightarrow(-y,x)$.
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Rotation $90^\circ$ counterclockwise about the origin. $(x,y) o (-y,x)$ rotates $90°$ counterclockwise.
Rotation $90^\circ$ counterclockwise about the origin. $(x,y) o (-y,x)$ rotates $90°$ counterclockwise.
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What is the coordinate rule for translating a point by $\langle a,b\rangle$?
What is the coordinate rule for translating a point by $\langle a,b\rangle$?
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$(x,y)\rightarrow(x+a,y+b)$. Add $a$ to $x$-coordinate and $b$ to $y$-coordinate.
$(x,y)\rightarrow(x+a,y+b)$. Add $a$ to $x$-coordinate and $b$ to $y$-coordinate.
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What does a rotation do to a figure relative to a fixed point?
What does a rotation do to a figure relative to a fixed point?
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Turns the figure around a center by a given angle and direction. All points maintain their distance from the center.
Turns the figure around a center by a given angle and direction. All points maintain their distance from the center.
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What does a reflection do to a figure relative to a line?
What does a reflection do to a figure relative to a line?
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Flips the figure across a line, creating a mirror image. Each point maps to its perpendicular distance on opposite side.
Flips the figure across a line, creating a mirror image. Each point maps to its perpendicular distance on opposite side.
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What is the coordinate rule for reflecting a point across the $x$-axis?
What is the coordinate rule for reflecting a point across the $x$-axis?
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$(x,y)\rightarrow(x,-y)$. Negate the $y$-coordinate while keeping $x$ unchanged.
$(x,y)\rightarrow(x,-y)$. Negate the $y$-coordinate while keeping $x$ unchanged.
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Which transformations are rigid motions that always preserve congruence?
Which transformations are rigid motions that always preserve congruence?
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Translations, rotations, and reflections. These three transformations preserve distances and angles.
Translations, rotations, and reflections. These three transformations preserve distances and angles.
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What property of distance is preserved by any rigid motion?
What property of distance is preserved by any rigid motion?
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All segment lengths (distances) stay the same. Rigid motions preserve the shape and size of figures.
All segment lengths (distances) stay the same. Rigid motions preserve the shape and size of figures.
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What property of angle measure is preserved by any rigid motion?
What property of angle measure is preserved by any rigid motion?
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All angle measures stay the same. Rigid motions don't change the shape of figures.
All angle measures stay the same. Rigid motions don't change the shape of figures.
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What does a translation do to every point of a figure on the coordinate plane?
What does a translation do to every point of a figure on the coordinate plane?
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Moves every point the same distance in the same direction. Translation slides without rotating or flipping.
Moves every point the same distance in the same direction. Translation slides without rotating or flipping.
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What is the coordinate rule for a $270^\circ$ counterclockwise rotation about the origin?
What is the coordinate rule for a $270^\circ$ counterclockwise rotation about the origin?
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$(x,y)\rightarrow(y,-x)$. Swap coordinates and negate the new $y$-coordinate.
$(x,y)\rightarrow(y,-x)$. Swap coordinates and negate the new $y$-coordinate.
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What is the coordinate rule for reflecting a point across the line $y=x$?
What is the coordinate rule for reflecting a point across the line $y=x$?
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$(x,y)\rightarrow(y,x)$. Swap coordinates to reflect across the diagonal.
$(x,y)\rightarrow(y,x)$. Swap coordinates to reflect across the diagonal.
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What is the coordinate rule for reflecting a point across the $y$-axis?
What is the coordinate rule for reflecting a point across the $y$-axis?
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$(x,y)\rightarrow(-x,y)$. Negate the $x$-coordinate while keeping $y$ unchanged.
$(x,y)\rightarrow(-x,y)$. Negate the $x$-coordinate while keeping $y$ unchanged.
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Which sequence maps $(x,y)$ to $(-x,y)$: reflect across $x$-axis or across $y$-axis?
Which sequence maps $(x,y)$ to $(-x,y)$: reflect across $x$-axis or across $y$-axis?
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Reflect across the $y$-axis. Changing $(x,y)$ to $(-x,y)$ negates only $x$-coordinate.
Reflect across the $y$-axis. Changing $(x,y)$ to $(-x,y)$ negates only $x$-coordinate.
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Find the image of $S(-6,4)$ after a $180^\circ$ rotation about the origin.
Find the image of $S(-6,4)$ after a $180^\circ$ rotation about the origin.
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$S'(6,-4)$. Apply rule $(x,y)\rightarrow(-x,-y)$ to get $(-(-6),-4)=(6,-4)$.
$S'(6,-4)$. Apply rule $(x,y)\rightarrow(-x,-y)$ to get $(-(-6),-4)=(6,-4)$.
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Find the image of $R(3,-2)$ after a $90^\circ$ counterclockwise rotation about the origin.
Find the image of $R(3,-2)$ after a $90^\circ$ counterclockwise rotation about the origin.
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$R'(2,3)$. Apply rule $(x,y)\rightarrow(-y,x)$ to get $(-(-2),3)=(2,3)$.
$R'(2,3)$. Apply rule $(x,y)\rightarrow(-y,x)$ to get $(-(-2),3)=(2,3)$.
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Find the image of $Q(-4,1)$ after reflection across the $y$-axis.
Find the image of $Q(-4,1)$ after reflection across the $y$-axis.
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$Q'(4,1)$. Reflecting across $y$-axis changes sign of $x$-coordinate.
$Q'(4,1)$. Reflecting across $y$-axis changes sign of $x$-coordinate.
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