Congruence via Rigid Motions - Geometry
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What is the coordinate rule for a $180^\circ$ rotation about the origin?
What is the coordinate rule for a $180^\circ$ rotation about the origin?
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$\left(x,y\right)\to\left(-x,-y\right)$. Half rotation flips both coordinates through the origin.
$\left(x,y\right)\to\left(-x,-y\right)$. Half rotation flips both coordinates through the origin.
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What is an invariant under any rigid motion?
What is an invariant under any rigid motion?
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Distance between any two points. Rigid motions preserve all distances by definition.
Distance between any two points. Rigid motions preserve all distances by definition.
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What is the definition of a rigid motion in the plane?
What is the definition of a rigid motion in the plane?
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A transformation that preserves all distances and angle measures. This is the fundamental property that defines rigid motions.
A transformation that preserves all distances and angle measures. This is the fundamental property that defines rigid motions.
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What does it mean for two figures to be congruent using rigid motions?
What does it mean for two figures to be congruent using rigid motions?
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One can be mapped onto the other by a sequence of rigid motions. Congruence is defined as the ability to map one figure onto another using rigid motions.
One can be mapped onto the other by a sequence of rigid motions. Congruence is defined as the ability to map one figure onto another using rigid motions.
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What three transformations are the basic rigid motions in the plane?
What three transformations are the basic rigid motions in the plane?
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Translations, rotations, and reflections. These are the only transformations that preserve distance and angle measure.
Translations, rotations, and reflections. These are the only transformations that preserve distance and angle measure.
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What is the coordinate rule for a translation by vector $\langle a,b\rangle$?
What is the coordinate rule for a translation by vector $\langle a,b\rangle$?
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$\left(x,y\right)\to\left(x+a,y+b\right)$. Add the components of the translation vector to the coordinates.
$\left(x,y\right)\to\left(x+a,y+b\right)$. Add the components of the translation vector to the coordinates.
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What is the coordinate rule for reflection across the $x$-axis?
What is the coordinate rule for reflection across the $x$-axis?
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$\left(x,y\right)\to\left(x,-y\right)$. The $x$-coordinate stays the same, the $y$-coordinate changes sign.
$\left(x,y\right)\to\left(x,-y\right)$. The $x$-coordinate stays the same, the $y$-coordinate changes sign.
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What is the coordinate rule for reflection across the $y$-axis?
What is the coordinate rule for reflection across the $y$-axis?
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$\left(x,y\right)\to\left(-x,y\right)$. The $y$-coordinate stays the same, the $x$-coordinate changes sign.
$\left(x,y\right)\to\left(-x,y\right)$. The $y$-coordinate stays the same, the $x$-coordinate changes sign.
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What is the coordinate rule for reflection across the line $y=x$?
What is the coordinate rule for reflection across the line $y=x$?
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$\left(x,y\right)\to\left(y,x\right)$. The coordinates swap positions across the diagonal line $y=x$.
$\left(x,y\right)\to\left(y,x\right)$. The coordinates swap positions across the diagonal line $y=x$.
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What is the coordinate rule for reflection across the line $y=-x$?
What is the coordinate rule for reflection across the line $y=-x$?
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$\left(x,y\right)\to\left(-y,-x\right)$. Both coordinates change sign and swap positions across $y=-x$.
$\left(x,y\right)\to\left(-y,-x\right)$. Both coordinates change sign and swap positions across $y=-x$.
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What is the coordinate rule for a $90^\circ$ counterclockwise rotation about the origin?
What is the coordinate rule for a $90^\circ$ counterclockwise rotation about the origin?
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$\left(x,y\right)\to\left(-y,x\right)$. Standard rotation matrix for $90°$ counterclockwise about the origin.
$\left(x,y\right)\to\left(-y,x\right)$. Standard rotation matrix for $90°$ counterclockwise about the origin.
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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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$\left(x,y\right)\to\left(y,-x\right)$. This is the $90°$ clockwise rotation transformation rule.
$\left(x,y\right)\to\left(y,-x\right)$. This is the $90°$ clockwise rotation transformation rule.
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What is the coordinate rule for reflection across the vertical line $x=a$?
What is the coordinate rule for reflection across the vertical line $x=a$?
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$\left(x,y\right)\to\left(2a-x,y\right)$. Reflects across a vertical line by using the distance formula.
$\left(x,y\right)\to\left(2a-x,y\right)$. Reflects across a vertical line by using the distance formula.
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What is the coordinate rule for reflection across the horizontal line $y=b$?
What is the coordinate rule for reflection across the horizontal line $y=b$?
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$\left(x,y\right)\to\left(x,2b-y\right)$. Reflects across a horizontal line by using the distance formula.
$\left(x,y\right)\to\left(x,2b-y\right)$. Reflects across a horizontal line by using the distance formula.
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What is the coordinate rule for a translation right $h$ and up $k$?
What is the coordinate rule for a translation right $h$ and up $k$?
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$\left(x,y\right)\to\left(x+h,y+k\right)$. Standard translation rule moving right $h$ units and up $k$ units.
$\left(x,y\right)\to\left(x+h,y+k\right)$. Standard translation rule moving right $h$ units and up $k$ units.
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Which rigid motion always preserves orientation (clockwise order of vertices)?
Which rigid motion always preserves orientation (clockwise order of vertices)?
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A translation or a rotation. These motions maintain the order of vertices around a figure.
A translation or a rotation. These motions maintain the order of vertices around a figure.
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Which rigid motion reverses orientation (clockwise becomes counterclockwise)?
Which rigid motion reverses orientation (clockwise becomes counterclockwise)?
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A reflection. Reflection flips the figure, reversing vertex order.
A reflection. Reflection flips the figure, reversing vertex order.
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What is the definition of the image of a point $P$ under a transformation?
What is the definition of the image of a point $P$ under a transformation?
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The point $P'$ obtained after applying the transformation to $P$. The image is the result after applying the transformation.
The point $P'$ obtained after applying the transformation to $P$. The image is the result after applying the transformation.
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What is an invariant under any rigid motion?
What is an invariant under any rigid motion?
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Distance between any two points. Rigid motions preserve all distances by definition.
Distance between any two points. Rigid motions preserve all distances by definition.
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What angle relationship is preserved under any rigid motion?
What angle relationship is preserved under any rigid motion?
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Angle measure is preserved. All angle measures remain the same under rigid motions.
Angle measure is preserved. All angle measures remain the same under rigid motions.
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What happens to the perimeter of a figure under a rigid motion?
What happens to the perimeter of a figure under a rigid motion?
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Perimeter is unchanged. Rigid motions preserve all lengths, so perimeter stays constant.
Perimeter is unchanged. Rigid motions preserve all lengths, so perimeter stays constant.
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What happens to the area of a figure under a rigid motion?
What happens to the area of a figure under a rigid motion?
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Area is unchanged. Rigid motions preserve shape and size, so area stays constant.
Area is unchanged. Rigid motions preserve shape and size, so area stays constant.
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What is the definition of the preimage in a transformation problem?
What is the definition of the preimage in a transformation problem?
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The original figure before the transformation. The preimage is the starting figure before transformation.
The original figure before the transformation. The preimage is the starting figure before transformation.
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What is the definition of the image in a transformation problem?
What is the definition of the image in a transformation problem?
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The resulting figure after the transformation. The image is the final figure after transformation.
The resulting figure after the transformation. The image is the final figure after transformation.
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Identify the rigid motion: $\left(x,y\right)\to\left(x+5,y-2\right)$.
Identify the rigid motion: $\left(x,y\right)\to\left(x+5,y-2\right)$.
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Translation by $\langle 5,-2\rangle$. Adding constants to coordinates indicates translation.
Translation by $\langle 5,-2\rangle$. Adding constants to coordinates indicates translation.
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Identify the rigid motion: $\left(x,y\right)\to\left(-x,y\right)$.
Identify the rigid motion: $\left(x,y\right)\to\left(-x,y\right)$.
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Reflection across the $y$-axis. The $x$-coordinate changes sign while $y$ remains unchanged.
Reflection across the $y$-axis. The $x$-coordinate changes sign while $y$ remains unchanged.
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Identify the rigid motion: $\left(x,y\right)\to\left(y,x\right)$.
Identify the rigid motion: $\left(x,y\right)\to\left(y,x\right)$.
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Reflection across the line $y=x$. Coordinates swap positions in this reflection transformation.
Reflection across the line $y=x$. Coordinates swap positions in this reflection transformation.
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Identify the rigid motion: $\left(x,y\right)\to\left(-y,x\right)$.
Identify the rigid motion: $\left(x,y\right)\to\left(-y,x\right)$.
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$90^\circ$ counterclockwise rotation about the origin. This transformation rotates points $90°$ counterclockwise.
$90^\circ$ counterclockwise rotation about the origin. This transformation rotates points $90°$ counterclockwise.
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Identify the rigid motion: $\left(x,y\right)\to\left(-x,-y\right)$.
Identify the rigid motion: $\left(x,y\right)\to\left(-x,-y\right)$.
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$180^\circ$ rotation about the origin. Both coordinates change sign for $180°$ rotation.
$180^\circ$ rotation about the origin. Both coordinates change sign for $180°$ rotation.
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What is the image of $\left(3,-4\right)$ after reflection across the $x$-axis?
What is the image of $\left(3,-4\right)$ after reflection across the $x$-axis?
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$\left(3,4\right)$. Reflection across $x$-axis changes $y$-coordinate sign.
$\left(3,4\right)$. Reflection across $x$-axis changes $y$-coordinate sign.
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