Performing and Sequencing Rigid Transformations - Geometry
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What is the coordinate rule for a $90^\circ$ clockwise rotation about the origin?
What is the coordinate rule for a $90^\circ$ clockwise rotation about the origin?
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$(x,y)\to(y,-x)$. Standard rotation matrix for $90°$ clockwise.
$(x,y)\to(y,-x)$. Standard rotation matrix for $90°$ clockwise.
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What is the image of triangle vertex $P(-2,3)$ after reflection across $x=0$?
What is the image of triangle vertex $P(-2,3)$ after reflection across $x=0$?
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$(2,3)$. Reflection across $y$-axis negates $x$-coordinate.
$(2,3)$. Reflection across $y$-axis negates $x$-coordinate.
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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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$(x,y)\to(x,-y)$. Negate the $y$-coordinate to flip across horizontal axis.
$(x,y)\to(x,-y)$. Negate the $y$-coordinate to flip across horizontal axis.
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What is the image of $(1,4)$ under a $90^\circ$ counterclockwise rotation about $(2,1)$?
What is the image of $(1,4)$ under a $90^\circ$ counterclockwise rotation about $(2,1)$?
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$(-1,0)$. Apply $90°$ CCW rotation formula about $(2,1)$.
$(-1,0)$. Apply $90°$ CCW rotation formula about $(2,1)$.
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Identify the transformation that maps $(x,y)$ to $(-y,-x)$.
Identify the transformation that maps $(x,y)$ to $(-y,-x)$.
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Reflection across $y=-x$. Negating both then swapping indicates reflection across $y=-x$.
Reflection across $y=-x$. Negating both then swapping indicates reflection across $y=-x$.
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What is the image of $(-3,5)$ under reflection across the line $y=2$?
What is the image of $(-3,5)$ under reflection across the line $y=2$?
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$(-3,-1)$. Apply rule $(x,2k-y)$ with $k=2$: $(-3,4-5) = (-3,-1)$.
$(-3,-1)$. Apply rule $(x,2k-y)$ with $k=2$: $(-3,4-5) = (-3,-1)$.
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What is the inverse transformation of translation by $\langle a,b\rangle$?
What is the inverse transformation of translation by $\langle a,b\rangle$?
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Translation by $\langle -a,-b\rangle$. Inverse translation uses opposite direction vector.
Translation by $\langle -a,-b\rangle$. Inverse translation uses opposite direction vector.
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What is the composition of two translations by $\langle a,b\rangle$ and $\langle c,d\rangle$?
What is the composition of two translations by $\langle a,b\rangle$ and $\langle c,d\rangle$?
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Translation by $\langle a+c,b+d\rangle$. Add corresponding components of translation vectors.
Translation by $\langle a+c,b+d\rangle$. Add corresponding components of translation vectors.
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What is the composition of reflections across $x$-axis then across $y=x$?
What is the composition of reflections across $x$-axis then across $y=x$?
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Rotation $90^\circ$ counterclockwise about the origin. Composition of these reflections produces counterclockwise rotation.
Rotation $90^\circ$ counterclockwise about the origin. Composition of these reflections produces counterclockwise rotation.
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What is the composition of reflections across $y=x$ then across the $x$-axis?
What is the composition of reflections across $y=x$ then across the $x$-axis?
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Rotation $90^\circ$ clockwise about the origin. Composition of these reflections produces clockwise rotation.
Rotation $90^\circ$ clockwise about the origin. Composition of these reflections produces clockwise rotation.
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What is the composition of reflections across the $x$-axis then the $y$-axis?
What is the composition of reflections across the $x$-axis then the $y$-axis?
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Rotation $180^\circ$ about the origin. Two reflections across perpendicular lines equals $180°$ rotation.
Rotation $180^\circ$ about the origin. Two reflections across perpendicular lines equals $180°$ rotation.
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Identify the transformation that maps $(x,y)$ to $(-x,-y)$.
Identify the transformation that maps $(x,y)$ to $(-x,-y)$.
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Rotation $180^\circ$ about the origin. Rule $(-x,-y)$ represents $180°$ rotation.
Rotation $180^\circ$ about the origin. Rule $(-x,-y)$ represents $180°$ rotation.
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Identify the transformation that maps $(x,y)$ to $(y,-x)$.
Identify the transformation that maps $(x,y)$ to $(y,-x)$.
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Rotation $90^\circ$ clockwise about the origin. Rule $(y,-x)$ represents $90°$ clockwise rotation.
Rotation $90^\circ$ clockwise about the origin. Rule $(y,-x)$ represents $90°$ clockwise rotation.
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Identify the transformation that maps $(x,y)$ to $(-y,x)$.
Identify the transformation that maps $(x,y)$ to $(-y,x)$.
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Rotation $90^\circ$ counterclockwise about the origin. Rule $(-y,x)$ represents $90°$ counterclockwise rotation.
Rotation $90^\circ$ counterclockwise about the origin. Rule $(-y,x)$ represents $90°$ counterclockwise rotation.
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What is the inverse of a reflection across a line $\ell$?
What is the inverse of a reflection across a line $\ell$?
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The same reflection across $\ell$. Reflections are their own inverse transformation.
The same reflection across $\ell$. Reflections are their own inverse transformation.
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What is the image of segment endpoints $A(1,2)$ and $B(4,-1)$ after $\langle 3,5\rangle$?
What is the image of segment endpoints $A(1,2)$ and $B(4,-1)$ after $\langle 3,5\rangle$?
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$A'(4,7),\ B'(7,4)$. Add translation vector to each endpoint.
$A'(4,7),\ B'(7,4)$. Add translation vector to each endpoint.
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Which rigid motions preserve orientation?
Which rigid motions preserve orientation?
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Translations and rotations. These transformations maintain clockwise vertex ordering.
Translations and rotations. These transformations maintain clockwise vertex ordering.
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Which rigid motion reverses orientation (clockwise order of vertices)?
Which rigid motion reverses orientation (clockwise order of vertices)?
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Reflection. Reflections flip orientation while preserving shape.
Reflection. Reflections flip orientation while preserving shape.
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Which transformation always preserves distances and angle measures?
Which transformation always preserves distances and angle measures?
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Any rigid motion (translation, reflection, or rotation). Rigid motions preserve all distance and angle measurements.
Any rigid motion (translation, reflection, or rotation). Rigid motions preserve all distance and angle measurements.
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What is the inverse of a $90^\circ$ counterclockwise rotation about the origin?
What is the inverse of a $90^\circ$ counterclockwise rotation about the origin?
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A $90^\circ$ clockwise rotation about the origin. Inverse rotation is opposite direction by same angle.
A $90^\circ$ clockwise rotation about the origin. Inverse rotation is opposite direction by same angle.
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What is the image of $(1,4)$ under a $90^\circ$ clockwise rotation about $(2,1)$?
What is the image of $(1,4)$ under a $90^\circ$ clockwise rotation about $(2,1)$?
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$(5,2)$. Apply $90°$ CW rotation formula about $(2,1)$.
$(5,2)$. Apply $90°$ CW rotation formula about $(2,1)$.
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Identify the transformation that maps $(x,y)$ to $(x+7,y-3)$.
Identify the transformation that maps $(x,y)$ to $(x+7,y-3)$.
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Translation by $\langle 7,-3\rangle$. Translation vector is $\langle 7,-3\rangle$ from coordinate rule.
Translation by $\langle 7,-3\rangle$. Translation vector is $\langle 7,-3\rangle$ from coordinate rule.
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Identify the transformation that maps $(x,y)$ to $(x,-y)$.
Identify the transformation that maps $(x,y)$ to $(x,-y)$.
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Reflection across the $x$-axis. Negating $y$-coordinate indicates reflection across $x$-axis.
Reflection across the $x$-axis. Negating $y$-coordinate indicates reflection across $x$-axis.
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What is the image of point $(0,-5)$ after reflection across $y=2$?
What is the image of point $(0,-5)$ after reflection across $y=2$?
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$(0,9)$. Apply reflection rule $(x,2k-y)$ with $k=2$.
$(0,9)$. Apply reflection rule $(x,2k-y)$ with $k=2$.
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What is the image of $(2,-3)$ after $90^\circ$ counterclockwise rotation about the origin?
What is the image of $(2,-3)$ after $90^\circ$ counterclockwise rotation about the origin?
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$(3,2)$. Apply CCW rotation rule $(-y,x)$: $(-(-3),2) = (3,2)$.
$(3,2)$. Apply CCW rotation rule $(-y,x)$: $(-(-3),2) = (3,2)$.
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What is the image of $(2,-3)$ after $90^\circ$ clockwise rotation about the origin?
What is the image of $(2,-3)$ after $90^\circ$ clockwise rotation about the origin?
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$(-3,-2)$. Apply CW rotation rule $(y,-x)$: $(-3,-2)$.
$(-3,-2)$. Apply CW rotation rule $(y,-x)$: $(-3,-2)$.
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What sequence maps $(x,y)$ to $(2-y,3+x)$ using a $90^\circ$ rotation and a translation?
What sequence maps $(x,y)$ to $(2-y,3+x)$ using a $90^\circ$ rotation and a translation?
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Rotate $90^\circ$ CCW about origin, then translate by $\langle 2,3\rangle$. First rotate, then translate by vector components.
Rotate $90^\circ$ CCW about origin, then translate by $\langle 2,3\rangle$. First rotate, then translate by vector components.
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Identify the transformation that maps $(x,y)$ to $(-x,y)$.
Identify the transformation that maps $(x,y)$ to $(-x,y)$.
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Reflection across the $y$-axis. Negating $x$-coordinate indicates reflection across $y$-axis.
Reflection across the $y$-axis. Negating $x$-coordinate indicates reflection across $y$-axis.
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What is the image of $(6,3)$ under a $180^\circ$ rotation about $(2,-1)$?
What is the image of $(6,3)$ under a $180^\circ$ rotation about $(2,-1)$?
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$(-2,-5)$. Apply rule $(2h-x,2k-y)$ with center $(2,-1)$.
$(-2,-5)$. Apply rule $(2h-x,2k-y)$ with center $(2,-1)$.
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Identify the transformation that maps $(x,y)$ to $(y,x)$.
Identify the transformation that maps $(x,y)$ to $(y,x)$.
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Reflection across $y=x$. Swapping coordinates indicates reflection across $y=x$.
Reflection across $y=x$. Swapping coordinates indicates reflection across $y=x$.
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