Defining Rotations, Reflections, and Translations - Geometry
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What points are fixed (unchanged) by a reflection across line $l$?
What points are fixed (unchanged) by a reflection across line $l$?
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Exactly the points on $l$. Points on the mirror line map to themselves.
Exactly the points on $l$. Points on the mirror line map to themselves.
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Which rigid motion is defined using perpendicular lines to a given line $l$?
Which rigid motion is defined using perpendicular lines to a given line $l$?
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A reflection across $l$. Each point reflects across line $l$ using perpendicular projection.
A reflection across $l$. Each point reflects across line $l$ using perpendicular projection.
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What geometric locus describes all points $P'$ that a rotation can send a given point $P$ to?
What geometric locus describes all points $P'$ that a rotation can send a given point $P$ to?
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A circle centered at $O$ with radius $OP$. All points equidistant from center trace circular paths.
A circle centered at $O$ with radius $OP$. All points equidistant from center trace circular paths.
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What is the definition of the image of point $P$ under reflection across line $l$?
What is the definition of the image of point $P$ under reflection across line $l$?
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$P'$ such that $l$ is the perpendicular bisector of segment $PP'$. The mirror line is the perpendicular bisector of segment $PP'$.
$P'$ such that $l$ is the perpendicular bisector of segment $PP'$. The mirror line is the perpendicular bisector of segment $PP'$.
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What is the sign convention for a positive rotation angle in the coordinate plane?
What is the sign convention for a positive rotation angle in the coordinate plane?
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Positive angles rotate counterclockwise. Standard mathematical convention for angle measurement.
Positive angles rotate counterclockwise. Standard mathematical convention for angle measurement.
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Which rigid motion is defined using parallel lines through each point?
Which rigid motion is defined using parallel lines through each point?
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A translation (motion along parallel directed segments). All points move along parallel directed line segments.
A translation (motion along parallel directed segments). All points move along parallel directed line segments.
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Which rigid motion is defined using perpendicular lines to a given line $l$?
Which rigid motion is defined using perpendicular lines to a given line $l$?
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A reflection across $l$. Each point reflects across line $l$ using perpendicular projection.
A reflection across $l$. Each point reflects across line $l$ using perpendicular projection.
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What is a translation in the plane, defined using directed line segments?
What is a translation in the plane, defined using directed line segments?
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A mapping that moves every point by the same directed segment (vector). Each point moves by the same displacement vector.
A mapping that moves every point by the same directed segment (vector). Each point moves by the same displacement vector.
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What is the defining distance property of a translation for any points $P$ and $Q$?
What is the defining distance property of a translation for any points $P$ and $Q$?
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$PQ=P'Q'$ (translations preserve distances). All distances between points remain unchanged.
$PQ=P'Q'$ (translations preserve distances). All distances between points remain unchanged.
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What points are fixed (unchanged) by a reflection across line $l$?
What points are fixed (unchanged) by a reflection across line $l$?
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Exactly the points on $l$. Points on the mirror line map to themselves.
Exactly the points on $l$. Points on the mirror line map to themselves.
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What point is fixed by any rotation about center $O$?
What point is fixed by any rotation about center $O$?
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The center $O$ is fixed. The center of rotation doesn't move during rotation.
The center $O$ is fixed. The center of rotation doesn't move during rotation.
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What is the fixed set of a translation by a nonzero vector?
What is the fixed set of a translation by a nonzero vector?
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No points are fixed (if the vector is nonzero). Every point moves by the same nonzero displacement.
No points are fixed (if the vector is nonzero). Every point moves by the same nonzero displacement.
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Identify the rigid motion: points move along circles centered at a fixed point.
Identify the rigid motion: points move along circles centered at a fixed point.
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Rotation. Points trace circular arcs around the center.
Rotation. Points trace circular arcs around the center.
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Identify the image of $(2,5)$ under reflection across the line $y=x$.
Identify the image of $(2,5)$ under reflection across the line $y=x$.
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$(5,2)$. Line $y=x$ reflection swaps coordinates.
$(5,2)$. Line $y=x$ reflection swaps coordinates.
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Which rigid motion is defined using circles centered at a fixed point $O$?
Which rigid motion is defined using circles centered at a fixed point $O$?
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A rotation about $O$. Points move along concentric circles around fixed center.
A rotation about $O$. Points move along concentric circles around fixed center.
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What does it mean to say a reflection preserves distance?
What does it mean to say a reflection preserves distance?
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For any $P,Q$, $PQ=P'Q'$ after reflection. All distances between points are preserved.
For any $P,Q$, $PQ=P'Q'$ after reflection. All distances between points are preserved.
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What does it mean to say a rotation preserves distance?
What does it mean to say a rotation preserves distance?
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For any $P,Q$, $PQ=P'Q'$ after rotation. Rotation is an isometry preserving all distances.
For any $P,Q$, $PQ=P'Q'$ after rotation. Rotation is an isometry preserving all distances.
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Identify the rigid motion: every point moves the same distance in the same direction.
Identify the rigid motion: every point moves the same distance in the same direction.
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Translation. Uniform motion without rotation or reflection.
Translation. Uniform motion without rotation or reflection.
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Identify the rigid motion: a line is the perpendicular bisector of each point-image segment.
Identify the rigid motion: a line is the perpendicular bisector of each point-image segment.
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Reflection. The mirror line bisects each point-image segment perpendicularly.
Reflection. The mirror line bisects each point-image segment perpendicularly.
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Identify the image of $(2,5)$ under reflection across the $x$-axis.
Identify the image of $(2,5)$ under reflection across the $x$-axis.
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$(2,-5)$. $x$-axis reflection negates the $y$-coordinate.
$(2,-5)$. $x$-axis reflection negates the $y$-coordinate.
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Identify the image of $(2,5)$ under reflection across the $y$-axis.
Identify the image of $(2,5)$ under reflection across the $y$-axis.
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$(-2,5)$. $y$-axis reflection negates the $x$-coordinate.
$(-2,5)$. $y$-axis reflection negates the $x$-coordinate.
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Identify the image of $(2,5)$ under reflection across the line $y=-x$.
Identify the image of $(2,5)$ under reflection across the line $y=-x$.
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$(-5,-2)$. Line $y=-x$ reflection swaps and negates coordinates.
$(-5,-2)$. Line $y=-x$ reflection swaps and negates coordinates.
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Identify the image of $(7,3)$ under reflection across the vertical line $x=2$.
Identify the image of $(7,3)$ under reflection across the vertical line $x=2$.
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$(-3,3)$. Reflect across $x=2$: $x$-coordinate becomes $2(2)-7=-3$.
$(-3,3)$. Reflect across $x=2$: $x$-coordinate becomes $2(2)-7=-3$.
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Identify the image of $(-1,6)$ under reflection across the horizontal line $y=4$.
Identify the image of $(-1,6)$ under reflection across the horizontal line $y=4$.
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$(-1,2)$. Reflect across $y=4$: $y$-coordinate becomes $2(4)-6=2$.
$(-1,2)$. Reflect across $y=4$: $y$-coordinate becomes $2(4)-6=2$.
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What is the center of rotation if $P$ maps to $P'$ and the center lies on the perpendicular bisector of $PP'$?
What is the center of rotation if $P$ maps to $P'$ and the center lies on the perpendicular bisector of $PP'$?
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The center is the intersection of perpendicular bisectors of two pairs $PP'$. Center equidistant from corresponding points in rotation.
The center is the intersection of perpendicular bisectors of two pairs $PP'$. Center equidistant from corresponding points in rotation.
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What line must contain the midpoint of $PP'$ for a reflection that maps $P$ to $P'$?
What line must contain the midpoint of $PP'$ for a reflection that maps $P$ to $P'$?
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The mirror line $l$ must pass through the midpoint of $PP'$. Mirror line is perpendicular bisector of segment $PP'$.
The mirror line $l$ must pass through the midpoint of $PP'$. Mirror line is perpendicular bisector of segment $PP'$.
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What must be true about $PP'$ if $P$ is reflected across line $l$ to $P'$?
What must be true about $PP'$ if $P$ is reflected across line $l$ to $P'$?
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Segment $PP'$ is perpendicular to $l$. Reflection requires perpendicularity to mirror line.
Segment $PP'$ is perpendicular to $l$. Reflection requires perpendicularity to mirror line.
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What must be true about distances to the mirror line in a reflection across $l$?
What must be true about distances to the mirror line in a reflection across $l$?
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$d(P,l)=d(P',l)$. Point and image are equidistant from mirror line.
$d(P,l)=d(P',l)$. Point and image are equidistant from mirror line.
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What is a translation in the plane, defined using directed line segments?
What is a translation in the plane, defined using directed line segments?
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A mapping that moves every point by the same directed segment (vector). Each point moves by the same displacement vector.
A mapping that moves every point by the same directed segment (vector). Each point moves by the same displacement vector.
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What point is fixed by any rotation about center $O$?
What point is fixed by any rotation about center $O$?
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The center $O$ is fixed. The center of rotation doesn't move during rotation.
The center $O$ is fixed. The center of rotation doesn't move during rotation.
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