Understand Similarity Through Transformations - 8th Grade Math
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What is the image of point $ (4,-6) $ after dilation about the origin with $ k=\frac{1}{2} $?
What is the image of point $ (4,-6) $ after dilation about the origin with $ k=\frac{1}{2} $?
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$ (2,-3) $. Apply dilation rule: $ (4 × \frac{1}{2}, -6 × \frac{1}{2}) = (2, -3) $
$ (2,-3) $. Apply dilation rule: $ (4 × \frac{1}{2}, -6 × \frac{1}{2}) = (2, -3) $
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What is the definition of scale factor in a dilation?
What is the definition of scale factor in a dilation?
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The ratio $ rac{ ext{image length}}{ ext{preimage length}}$. Compares new length to original length after dilation.
The ratio $ rac{ ext{image length}}{ ext{preimage length}}$. Compares new length to original length after dilation.
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What is true about side lengths in similar figures with scale factor $k$?
What is true about side lengths in similar figures with scale factor $k$?
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Corresponding side lengths have ratio $k$. Dilation multiplies all lengths by the scale factor.
Corresponding side lengths have ratio $k$. Dilation multiplies all lengths by the scale factor.
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Which option best describes a valid similarity sequence from figure $A$ to $B$ when $B$ is a rotated and enlarged copy?
Which option best describes a valid similarity sequence from figure $A$ to $B$ when $B$ is a rotated and enlarged copy?
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Rotate, then dilate (translation may be included to reposition). Order matters: rotate first to align, then scale up.
Rotate, then dilate (translation may be included to reposition). Order matters: rotate first to align, then scale up.
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What is the image of point $(2,7)$ after a $90^\circ$ counterclockwise rotation about the origin?
What is the image of point $(2,7)$ after a $90^\circ$ counterclockwise rotation about the origin?
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$(-7,2)$. Apply rotation rule: $(2, 7) → (-7, 2)$.
$(-7,2)$. Apply rotation rule: $(2, 7) → (-7, 2)$.
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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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$(x,y)\rightarrow(-y,x)$. 90° CCW rotation swaps coordinates and negates new $x$.
$(x,y)\rightarrow(-y,x)$. 90° CCW rotation swaps coordinates and negates new $x$.
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What is the coordinate rule for a reflection across the $y$-axis?
What is the coordinate rule for a reflection across the $y$-axis?
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$(x,y)\rightarrow(-x,y)$. Reflection across $y$-axis negates the $x$-coordinate.
$(x,y)\rightarrow(-x,y)$. Reflection across $y$-axis negates the $x$-coordinate.
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What is the coordinate rule for a reflection across the $x$-axis?
What is the coordinate rule for a reflection across the $x$-axis?
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$(x,y)\rightarrow(x,-y)$. Reflection across $x$-axis negates the $y$-coordinate.
$(x,y)\rightarrow(x,-y)$. Reflection across $x$-axis negates the $y$-coordinate.
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What is the image of point $(3,-1)$ after a translation by $(5,4)$?
What is the image of point $(3,-1)$ after a translation by $(5,4)$?
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$(8,3)$. Add translation vector: $(3+5, -1+4) = (8, 3)$.
$(8,3)$. Add translation vector: $(3+5, -1+4) = (8, 3)$.
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What is the coordinate rule for a translation by $(a,b)$?
What is the coordinate rule for a translation by $(a,b)$?
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$(x,y)\rightarrow(x+a,y+b)$. Add translation vector to each coordinate.
$(x,y)\rightarrow(x+a,y+b)$. Add translation vector to each coordinate.
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What is the image of point $(-2,5)$ after dilation about the origin with $k=3$?
What is the image of point $(-2,5)$ after dilation about the origin with $k=3$?
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$(-6,15)$. Apply dilation rule: $(-2×3, 5×3) = (-6, 15)$.
$(-6,15)$. Apply dilation rule: $(-2×3, 5×3) = (-6, 15)$.
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What is the coordinate rule for a dilation about the origin with scale factor $k$?
What is the coordinate rule for a dilation about the origin with scale factor $k$?
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$(x,y)\rightarrow(kx,ky)$. Each coordinate is multiplied by the scale factor.
$(x,y)\rightarrow(kx,ky)$. Each coordinate is multiplied by the scale factor.
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Identify the missing original side: if $k= rac{3}{4}$ and the image side is $9$, what was the original side?
Identify the missing original side: if $k= rac{3}{4}$ and the image side is $9$, what was the original side?
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$12$. Divide image by scale factor: $9 ÷ rac{3}{4} = 12$.
$12$. Divide image by scale factor: $9 ÷ rac{3}{4} = 12$.
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Identify the missing side: if $k=2$ and an original side is $7$, what is the image side length?
Identify the missing side: if $k=2$ and an original side is $7$, what is the image side length?
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$14$. Multiply original length by scale factor: $7 × 2 = 14$.
$14$. Multiply original length by scale factor: $7 × 2 = 14$.
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What is the scale factor from a figure with side $10$ to a similar figure with corresponding side $4$?
What is the scale factor from a figure with side $10$ to a similar figure with corresponding side $4$?
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$k= rac{4}{10}= rac{2}{5}$. Scale factor = image length ÷ original length.
$k= rac{4}{10}= rac{2}{5}$. Scale factor = image length ÷ original length.
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What is the scale factor from a figure with side $6$ to a similar figure with corresponding side $9$?
What is the scale factor from a figure with side $6$ to a similar figure with corresponding side $9$?
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$k= rac{9}{6}= rac{3}{2}$. Scale factor = image length ÷ original length.
$k= rac{9}{6}= rac{3}{2}$. Scale factor = image length ÷ original length.
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What is true about angle measures in similar figures after transformations and dilation?
What is true about angle measures in similar figures after transformations and dilation?
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All corresponding angles are equal. Dilations and rigid motions preserve angle measures.
All corresponding angles are equal. Dilations and rigid motions preserve angle measures.
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What does a dilation with scale factor $k$ do to all lengths in a figure?
What does a dilation with scale factor $k$ do to all lengths in a figure?
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It multiplies every length by $k$. Dilation scales all distances from center by the same factor.
It multiplies every length by $k$. Dilation scales all distances from center by the same factor.
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Which transformations are rigid motions (do not change size): rotation, reflection, translation, dilation?
Which transformations are rigid motions (do not change size): rotation, reflection, translation, dilation?
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Rotation, reflection, and translation. These preserve distances and angles; dilation changes size.
Rotation, reflection, and translation. These preserve distances and angles; dilation changes size.
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What does it mean for two $2$-D figures to be similar using transformations?
What does it mean for two $2$-D figures to be similar using transformations?
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One can be mapped to the other by rigid motions and a dilation. Rigid motions preserve shape; dilation changes size proportionally.
One can be mapped to the other by rigid motions and a dilation. Rigid motions preserve shape; dilation changes size proportionally.
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What is the image of $Q(-4,1)$ after a $90^\circ$ counterclockwise rotation about the origin?
What is the image of $Q(-4,1)$ after a $90^\circ$ counterclockwise rotation about the origin?
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$Q'(-1,-4)$. 90° CCW rotation: $(x,y) \rightarrow (-y,x)$.
$Q'(-1,-4)$. 90° CCW rotation: $(x,y) \rightarrow (-y,x)$.
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Which sequence maps $A(0,0)$ to $A'(4,-1)$ using one rigid motion?
Which sequence maps $A(0,0)$ to $A'(4,-1)$ using one rigid motion?
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Translate right $4$ and down $1$. Add 4 to x-coordinate and subtract 1 from y-coordinate.
Translate right $4$ and down $1$. Add 4 to x-coordinate and subtract 1 from y-coordinate.
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Identify whether rectangles with sides $3,5$ and $6,10$ are similar.
Identify whether rectangles with sides $3,5$ and $6,10$ are similar.
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Yes, because $\frac{6}{3}=\frac{10}{5}=2$. Both ratios equal 2, so sides are proportional.
Yes, because $\frac{6}{3}=\frac{10}{5}=2$. Both ratios equal 2, so sides are proportional.
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Identify whether the figures are similar if angle measures are $40^\circ,60^\circ,80^\circ$ and $40^\circ,60^\circ,80^\circ$.
Identify whether the figures are similar if angle measures are $40^\circ,60^\circ,80^\circ$ and $40^\circ,60^\circ,80^\circ$.
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Yes, the angles match (AAA similarity). Same angles guarantee similarity for triangles.
Yes, the angles match (AAA similarity). Same angles guarantee similarity for triangles.
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What is the original length if the image length is $14$ under dilation with $k=2$?
What is the original length if the image length is $14$ under dilation with $k=2$?
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$7$. Divide image length by scale factor: $14 \div 2 = 7$.
$7$. Divide image length by scale factor: $14 \div 2 = 7$.
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