AA Criterion from Similarity Transformations - Geometry
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What is the similarity ratio between perimeters of similar figures with scale factor $k$?
What is the similarity ratio between perimeters of similar figures with scale factor $k$?
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$\frac{P_2}{P_1}=k$. Perimeters scale by the same factor as corresponding sides.
$\frac{P_2}{P_1}=k$. Perimeters scale by the same factor as corresponding sides.
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What is the similarity ratio between areas of similar figures with scale factor $k$?
What is the similarity ratio between areas of similar figures with scale factor $k$?
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$\frac{A_2}{A_1}=k^2$. Areas scale by the square of the side scale factor.
$\frac{A_2}{A_1}=k^2$. Areas scale by the square of the side scale factor.
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What must be true about corresponding side lengths if two figures are similar?
What must be true about corresponding side lengths if two figures are similar?
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All corresponding side ratios must equal the same constant $k$. This ensures proportional scaling in all directions.
All corresponding side ratios must equal the same constant $k$. This ensures proportional scaling in all directions.
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Find the perimeter scale factor if corresponding side scale factor is $k=\frac{4}{5}$.
Find the perimeter scale factor if corresponding side scale factor is $k=\frac{4}{5}$.
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$\frac{4}{5}$. Perimeters scale by the same factor as corresponding sides.
$\frac{4}{5}$. Perimeters scale by the same factor as corresponding sides.
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Which transformations preserve distances exactly (not just proportionally)?
Which transformations preserve distances exactly (not just proportionally)?
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Rigid motions: translations, rotations, and reflections. These transformations preserve all distances exactly.
Rigid motions: translations, rotations, and reflections. These transformations preserve all distances exactly.
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What is the effect of a dilation on angles?
What is the effect of a dilation on angles?
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Angles are preserved (remain congruent). Dilation only changes size, not angle measures.
Angles are preserved (remain congruent). Dilation only changes size, not angle measures.
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Identify the missing ratio if $\triangle ABC\sim\triangle DEF$ and $\frac{AB}{DE}=\frac{2}{3}$.
Identify the missing ratio if $\triangle ABC\sim\triangle DEF$ and $\frac{AB}{DE}=\frac{2}{3}$.
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$\frac{BC}{EF}=\frac{2}{3}$. All corresponding side ratios must be equal in similar triangles.
$\frac{BC}{EF}=\frac{2}{3}$. All corresponding side ratios must be equal in similar triangles.
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What is the effect of a dilation with $k>1$ on a figure?
What is the effect of a dilation with $k>1$ on a figure?
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It enlarges the figure by factor $k$. Scale factor greater than 1 increases all dimensions.
It enlarges the figure by factor $k$. Scale factor greater than 1 increases all dimensions.
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Identify whether triangles are similar if one has angles $30^\circ,60^\circ,90^\circ$ and the other $30^\circ,50^\circ,100^\circ$.
Identify whether triangles are similar if one has angles $30^\circ,60^\circ,90^\circ$ and the other $30^\circ,50^\circ,100^\circ$.
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No; corresponding angles are not congruent. Different angle measures mean the triangles are not similar.
No; corresponding angles are not congruent. Different angle measures mean the triangles are not similar.
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What is the minimum transformation set needed to describe similarity between two figures?
What is the minimum transformation set needed to describe similarity between two figures?
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A dilation and zero or more rigid motions. Dilation handles size changes; rigid motions align position and orientation.
A dilation and zero or more rigid motions. Dilation handles size changes; rigid motions align position and orientation.
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Identify the scale factor if corresponding sides are $14$ (image) and $-7$ (preimage direction ignored).
Identify the scale factor if corresponding sides are $14$ (image) and $-7$ (preimage direction ignored).
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$|k|=\frac{14}{7}=2$. Use absolute value since direction doesn't affect scale magnitude.
$|k|=\frac{14}{7}=2$. Use absolute value since direction doesn't affect scale magnitude.
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Find the image perimeter if $P_1=36$ and the dilation scale factor is $k=\frac{5}{6}$.
Find the image perimeter if $P_1=36$ and the dilation scale factor is $k=\frac{5}{6}$.
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$P_2=36\cdot\frac{5}{6}=30$. Multiply the original perimeter by the scale factor.
$P_2=36\cdot\frac{5}{6}=30$. Multiply the original perimeter by the scale factor.
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Find the scale factor $k$ if similar triangles have areas in ratio $\frac{A_2}{A_1}=\frac{25}{4}$.
Find the scale factor $k$ if similar triangles have areas in ratio $\frac{A_2}{A_1}=\frac{25}{4}$.
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$k=\frac{5}{2}$. Take the square root of the area ratio: $\sqrt{\frac{25}{4}} = \frac{5}{2}$.
$k=\frac{5}{2}$. Take the square root of the area ratio: $\sqrt{\frac{25}{4}} = \frac{5}{2}$.
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Identify whether two triangles are similar if their corresponding angles are $40^\circ,60^\circ,80^\circ$ in both.
Identify whether two triangles are similar if their corresponding angles are $40^\circ,60^\circ,80^\circ$ in both.
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Yes; all corresponding angles are congruent. Identical corresponding angles guarantee triangle similarity (AAA).
Yes; all corresponding angles are congruent. Identical corresponding angles guarantee triangle similarity (AAA).
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Identify whether matching only one side ratio proves two triangles are similar.
Identify whether matching only one side ratio proves two triangles are similar.
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No; all corresponding side ratios must match (or enough angle information). Similarity requires all corresponding ratios to be equal.
No; all corresponding side ratios must match (or enough angle information). Similarity requires all corresponding ratios to be equal.
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What is the definition of similarity using similarity transformations?
What is the definition of similarity using similarity transformations?
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A sequence of rigid motions and a dilation maps one figure to the other. This is the formal definition of similar figures using transformations.
A sequence of rigid motions and a dilation maps one figure to the other. This is the formal definition of similar figures using transformations.
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What transformations are allowed in a similarity transformation sequence?
What transformations are allowed in a similarity transformation sequence?
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Rigid motions (translate, rotate, reflect) and dilations. These preserve shape and allow controlled size changes through dilation.
Rigid motions (translate, rotate, reflect) and dilations. These preserve shape and allow controlled size changes through dilation.
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What property of angle measures is preserved by any similarity transformation?
What property of angle measures is preserved by any similarity transformation?
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All angle measures are preserved (corresponding angles are congruent). Both rigid motions and dilations preserve angle measures.
All angle measures are preserved (corresponding angles are congruent). Both rigid motions and dilations preserve angle measures.
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What property of lengths is preserved by a dilation with scale factor $k$?
What property of lengths is preserved by a dilation with scale factor $k$?
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All lengths are multiplied by $k$. Dilation uniformly scales all distances by the same factor $k$.
All lengths are multiplied by $k$. Dilation uniformly scales all distances by the same factor $k$.
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What is the scale factor $k$ between similar figures?
What is the scale factor $k$ between similar figures?
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$k=\frac{\text{image length}}{\text{preimage length}}$ for any pair of corresponding sides. The ratio of any corresponding lengths gives the constant scale factor.
$k=\frac{\text{image length}}{\text{preimage length}}$ for any pair of corresponding sides. The ratio of any corresponding lengths gives the constant scale factor.
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What does it mean for two triangles to be similar in terms of angles?
What does it mean for two triangles to be similar in terms of angles?
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All corresponding angles are congruent. Angle measures are preserved under similarity transformations.
All corresponding angles are congruent. Angle measures are preserved under similarity transformations.
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What does it mean for two triangles to be similar in terms of side lengths?
What does it mean for two triangles to be similar in terms of side lengths?
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All corresponding side lengths are proportional with a common ratio $k$. All corresponding sides have the same ratio $k$.
All corresponding side lengths are proportional with a common ratio $k$. All corresponding sides have the same ratio $k$.
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What is the meaning of $\triangle ABC\sim\triangle DEF$ about correspondence?
What is the meaning of $\triangle ABC\sim\triangle DEF$ about correspondence?
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$A\leftrightarrow D$, $B\leftrightarrow E$, $C\leftrightarrow F$. The order of vertices indicates which angles and sides correspond.
$A\leftrightarrow D$, $B\leftrightarrow E$, $C\leftrightarrow F$. The order of vertices indicates which angles and sides correspond.
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What is the side proportion statement for $\triangle ABC\sim\triangle DEF$?
What is the side proportion statement for $\triangle ABC\sim\triangle DEF$?
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$\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$. Each ratio equals the common scale factor $k$.
$\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$. Each ratio equals the common scale factor $k$.
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What is the angle congruence statement for $\triangle ABC\sim\triangle DEF$?
What is the angle congruence statement for $\triangle ABC\sim\triangle DEF$?
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$\angle A\cong\angle D$, $\angle B\cong\angle E$, $\angle C\cong\angle F$. Corresponding angles are equal when triangles are similar.
$\angle A\cong\angle D$, $\angle B\cong\angle E$, $\angle C\cong\angle F$. Corresponding angles are equal when triangles are similar.
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Which transformations preserve distances exactly (not just proportionally)?
Which transformations preserve distances exactly (not just proportionally)?
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Rigid motions: translations, rotations, and reflections. These transformations preserve all distances exactly.
Rigid motions: translations, rotations, and reflections. These transformations preserve all distances exactly.
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What is the effect of a rigid motion on a figure’s size and shape?
What is the effect of a rigid motion on a figure’s size and shape?
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Size and shape are unchanged; all distances and angles are preserved. Rigid motions create congruent figures with identical measurements.
Size and shape are unchanged; all distances and angles are preserved. Rigid motions create congruent figures with identical measurements.
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What is the effect of a dilation on angles?
What is the effect of a dilation on angles?
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Angles are preserved (remain congruent). Dilation only changes size, not angle measures.
Angles are preserved (remain congruent). Dilation only changes size, not angle measures.
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What is the effect of a dilation with $k>1$ on a figure?
What is the effect of a dilation with $k>1$ on a figure?
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It enlarges the figure by factor $k$. Scale factor greater than 1 increases all dimensions.
It enlarges the figure by factor $k$. Scale factor greater than 1 increases all dimensions.
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What is the effect of a dilation with $0<k<1$ on a figure?
What is the effect of a dilation with $0<k<1$ on a figure?
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It reduces the figure by factor $k$. Scale factor between 0 and 1 decreases all dimensions.
It reduces the figure by factor $k$. Scale factor between 0 and 1 decreases all dimensions.
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