Transformations and Symmetry - ISEE Middle Level: Quantitative Reasoning
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What is the transformation rule for reflecting a point $(x,y)$ across the $x$-axis?
What is the transformation rule for reflecting a point $(x,y)$ across the $x$-axis?
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$(x,y)\rightarrow(x,-y)$. Reflection across the x-axis negates the y-coordinate while preserving the x-coordinate to mirror the point below or above the axis.
$(x,y)\rightarrow(x,-y)$. Reflection across the x-axis negates the y-coordinate while preserving the x-coordinate to mirror the point below or above the axis.
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What is the transformation rule for reflecting a point $(x,y)$ across the $y$-axis?
What is the transformation rule for reflecting a point $(x,y)$ across the $y$-axis?
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$(x,y)\rightarrow(-x,y)$. Reflection across the y-axis negates the x-coordinate while preserving the y-coordinate to mirror the point left or right of the axis.
$(x,y)\rightarrow(-x,y)$. Reflection across the y-axis negates the x-coordinate while preserving the y-coordinate to mirror the point left or right of the axis.
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What is the transformation rule for reflecting a point $(x,y)$ across the line $y=x$?
What is the transformation rule for reflecting a point $(x,y)$ across the line $y=x$?
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$(x,y)\rightarrow(y,x)$. Reflection across the line $y=x$ swaps the x- and y-coordinates to mirror the point over the diagonal.
$(x,y)\rightarrow(y,x)$. Reflection across the line $y=x$ swaps the x- and y-coordinates to mirror the point over the diagonal.
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What is the transformation rule for reflecting a point $(x,y)$ across the line $y=-x$?
What is the transformation rule for reflecting a point $(x,y)$ across the line $y=-x$?
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$(x,y)\rightarrow(-y,-x)$. Reflection across the line $y=-x$ swaps the coordinates and negates both to mirror the point over the anti-diagonal.
$(x,y)\rightarrow(-y,-x)$. Reflection across the line $y=-x$ swaps the coordinates and negates both to mirror the point over the anti-diagonal.
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What is the transformation rule for a $90^\circ$ counterclockwise rotation about the origin?
What is the transformation rule for a $90^\circ$ counterclockwise rotation about the origin?
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$(x,y)\rightarrow(-y,x)$. A $90^\circ$ counterclockwise rotation transforms the coordinates by setting new x to -y and new y to x.
$(x,y)\rightarrow(-y,x)$. A $90^\circ$ counterclockwise rotation transforms the coordinates by setting new x to -y and new y to x.
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What is the transformation rule for a $90^\circ$ clockwise rotation about the origin?
What is the transformation rule for a $90^\circ$ clockwise rotation about the origin?
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$(x,y)\rightarrow(y,-x)$. A $90^\circ$ clockwise rotation transforms the coordinates by setting new x to y and new y to -x.
$(x,y)\rightarrow(y,-x)$. A $90^\circ$ clockwise rotation transforms the coordinates by setting new x to y and new y to -x.
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What is the transformation rule for a $180^\circ$ rotation about the origin?
What is the transformation rule for a $180^\circ$ rotation about the origin?
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$(x,y)\rightarrow(-x,-y)$. A $180^\circ$ rotation negates both x- and y-coordinates to rotate the point halfway around the origin.
$(x,y)\rightarrow(-x,-y)$. A $180^\circ$ rotation negates both x- and y-coordinates to rotate the point halfway around the origin.
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What is the transformation rule for translating a point $(x,y)$ by $(a,b)$?
What is the transformation rule for translating a point $(x,y)$ by $(a,b)$?
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$(x,y)\rightarrow(x+a,y+b)$. Translation shifts the point by adding a to the x-coordinate and b to the y-coordinate.
$(x,y)\rightarrow(x+a,y+b)$. Translation shifts the point by adding a to the x-coordinate and b to the y-coordinate.
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What is the transformation rule for dilating a point $(x,y)$ by scale factor $k$ about the origin?
What is the transformation rule for dilating a point $(x,y)$ by scale factor $k$ about the origin?
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$(x,y)\rightarrow(kx,ky)$. Dilation enlarges or reduces the figure by multiplying both coordinates by the scale factor k from the origin.
$(x,y)\rightarrow(kx,ky)$. Dilation enlarges or reduces the figure by multiplying both coordinates by the scale factor k from the origin.
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What is the scale factor $k$ for a dilation if a segment of length $L$ becomes length $L'$?
What is the scale factor $k$ for a dilation if a segment of length $L$ becomes length $L'$?
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$k=\frac{L'}{L}$. The scale factor is the ratio of the image length to the original length under dilation.
$k=\frac{L'}{L}$. The scale factor is the ratio of the image length to the original length under dilation.
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Which transformations always preserve distance and angle measure: translation, rotation, reflection, or dilation?
Which transformations always preserve distance and angle measure: translation, rotation, reflection, or dilation?
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Translation, rotation, and reflection. These are rigid motions that preserve congruence, maintaining distances and angles, unlike dilation which alters size.
Translation, rotation, and reflection. These are rigid motions that preserve congruence, maintaining distances and angles, unlike dilation which alters size.
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What happens to area under a dilation with scale factor $k$ (in terms of the original area $A$)?
What happens to area under a dilation with scale factor $k$ (in terms of the original area $A$)?
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New area $=k^2A$. Areas scale by the square of the linear scale factor under dilation.
New area $=k^2A$. Areas scale by the square of the linear scale factor under dilation.
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Identify the type of symmetry: a figure can be mapped onto itself by a reflection across a line.
Identify the type of symmetry: a figure can be mapped onto itself by a reflection across a line.
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Line symmetry (reflectional symmetry). This symmetry allows the figure to coincide with itself when folded along the line of reflection.
Line symmetry (reflectional symmetry). This symmetry allows the figure to coincide with itself when folded along the line of reflection.
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Identify the type of symmetry: a figure can be mapped onto itself by a rotation about a point.
Identify the type of symmetry: a figure can be mapped onto itself by a rotation about a point.
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Rotational symmetry. This symmetry allows the figure to coincide with itself after rotation by a specific angle around a center point.
Rotational symmetry. This symmetry allows the figure to coincide with itself after rotation by a specific angle around a center point.
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What is the order of rotational symmetry if the smallest rotation that maps a figure onto itself is $120^\circ$?
What is the order of rotational symmetry if the smallest rotation that maps a figure onto itself is $120^\circ$?
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$3$. The order is $360^\circ$ divided by the smallest rotation angle that maps the figure onto itself.
$3$. The order is $360^\circ$ divided by the smallest rotation angle that maps the figure onto itself.
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What is the smallest positive rotation angle if a figure has rotational symmetry of order $5$?
What is the smallest positive rotation angle if a figure has rotational symmetry of order $5$?
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$72^\circ$. The smallest angle is $360^\circ$ divided by the order of rotational symmetry.
$72^\circ$. The smallest angle is $360^\circ$ divided by the order of rotational symmetry.
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Find the image of point $A(3,-5)$ after reflection across the $x$-axis.
Find the image of point $A(3,-5)$ after reflection across the $x$-axis.
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$(3,5)$. Apply the x-axis reflection rule by negating the y-coordinate of (3,-5).
$(3,5)$. Apply the x-axis reflection rule by negating the y-coordinate of (3,-5).
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Find the image of point $B(-4,2)$ after reflection across the $y$-axis.
Find the image of point $B(-4,2)$ after reflection across the $y$-axis.
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$(4,2)$. Apply the y-axis reflection rule by negating the x-coordinate of (-4,2).
$(4,2)$. Apply the y-axis reflection rule by negating the x-coordinate of (-4,2).
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Find the image of point $C(7,-1)$ after reflection across the line $y=x$.
Find the image of point $C(7,-1)$ after reflection across the line $y=x$.
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$(-1,7)$. Apply the $y=x$ reflection rule by swapping the coordinates of (7,-1).
$(-1,7)$. Apply the $y=x$ reflection rule by swapping the coordinates of (7,-1).
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Find the image of point $D(2,9)$ after a $90^\circ$ counterclockwise rotation about the origin.
Find the image of point $D(2,9)$ after a $90^\circ$ counterclockwise rotation about the origin.
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$(-9,2)$. Apply the $90^\circ$ counterclockwise rotation rule to (2,9) by setting new x to -9 and new y to 2.
$(-9,2)$. Apply the $90^\circ$ counterclockwise rotation rule to (2,9) by setting new x to -9 and new y to 2.
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Find the image of point $E(-6,1)$ after a $180^\circ$ rotation about the origin.
Find the image of point $E(-6,1)$ after a $180^\circ$ rotation about the origin.
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$(6,-1)$. Apply the $180^\circ$ rotation rule to (-6,1) by negating both coordinates.
$(6,-1)$. Apply the $180^\circ$ rotation rule to (-6,1) by negating both coordinates.
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Find the image of point $F(1,-3)$ after translation by $(4,-2)$.
Find the image of point $F(1,-3)$ after translation by $(4,-2)$.
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$(5,-5)$. Apply the translation by adding 4 to x and -2 to y of (1,-3).
$(5,-5)$. Apply the translation by adding 4 to x and -2 to y of (1,-3).
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A segment of length $8$ is dilated by scale factor $\frac{3}{2}$. What is the new length?
A segment of length $8$ is dilated by scale factor $\frac{3}{2}$. What is the new length?
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$12$. The new length is the original length multiplied by the scale factor $\frac{3}{2}$.
$12$. The new length is the original length multiplied by the scale factor $\frac{3}{2}$.
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A rectangle has area $10$. It is dilated by scale factor $3$. What is the new area?
A rectangle has area $10$. It is dilated by scale factor $3$. What is the new area?
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$90$. The new area is the original area multiplied by the square of the scale factor 3.
$90$. The new area is the original area multiplied by the square of the scale factor 3.
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A figure has line symmetry about the $y$-axis. If it contains $(5,-2)$, what reflected point must it contain?
A figure has line symmetry about the $y$-axis. If it contains $(5,-2)$, what reflected point must it contain?
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$(-5,-2)$. Symmetry about the y-axis requires the point symmetric to (5,-2) by negating its x-coordinate.
$(-5,-2)$. Symmetry about the y-axis requires the point symmetric to (5,-2) by negating its x-coordinate.
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