AA Criterion from Similarity Transformations - Geometry

1. Find the area scale factor if corresponding side scale factor is $k=3$.

   $9$. Area scales by the square of the side scale factor: $3^2 = 9$.

2. Find the scale factor $k$ if similar triangles have areas in ratio $\frac{A_2}{A_1}=\frac{25}{4}$.

   $k=\frac{5}{2}$. Take the square root of the area ratio: $\sqrt{\frac{25}{4}} = \frac{5}{2}$.

3. Find the image length of a segment of length $10$ under dilation with $k=\frac{3}{2}$.

   $15$. Multiply the original length by the scale factor.

4. What is a quick check for similarity of triangles using measures?

   Check congruent corresponding angles and proportional corresponding sides. These conditions together define triangle similarity.

5. What does it mean for two triangles to be similar in terms of side lengths?

   All corresponding side lengths are proportional with a common ratio $k$. All corresponding sides have the same ratio $k$.

6. What must be true about corresponding side lengths if two figures are similar?

   All corresponding side ratios must equal the same constant $k$. This ensures proportional scaling in all directions.

7. Identify whether triangles are similar if one has angles $30^\circ,60^\circ,90^\circ$ and the other $30^\circ,50^\circ,100^\circ$.

   No; corresponding angles are not congruent. Different angle measures mean the triangles are not similar.

8. Identify the scale factor $k$ if a side of length $6$ maps to a side of length $15$.

   $k=\frac{15}{6}=\frac{5}{2}$. Scale factor is image length divided by preimage length.

9. What is the minimum transformation set needed to describe similarity between two figures?

   A dilation and zero or more rigid motions. Dilation handles size changes; rigid motions align position and orientation.

10. Find $EF$ if $\triangle ABC\sim\triangle DEF$, $BC=12$, and $k=\frac{3}{2}$ from $ABC$ to $DEF$.

    $EF=18$. Multiply the preimage side by the scale factor: $12 \times \frac{3}{2} = 18$.

11. What is the key difference between congruence and similarity transformations?

    Similarity allows dilations; congruence uses only rigid motions. Congruence is similarity with scale factor exactly 1.

12. Identify whether a dilation can map a triangle to a congruent triangle when $k\ne 1$.

    No; congruence requires $k=1$. Dilation with $k \ne 1$ changes size, preventing congruence.

13. Identify $k$ from $\triangle DEF$ to $\triangle ABC$ if $k$ from $ABC$ to $DEF$ is $\frac{7}{3}$.

    $\frac{3}{7}$. Reciprocal of the original scale factor.

14. Identify whether matching only one side ratio proves two triangles are similar.

    No; all corresponding side ratios must match (or enough angle information). Similarity requires all corresponding ratios to be equal.

15. What is the similarity ratio between perimeters of similar figures with scale factor $k$?

    $\frac{P_2}{P_1}=k$. Perimeters scale by the same factor as corresponding sides.

16. What property of angle measures is preserved by any similarity transformation?

    All angle measures are preserved (corresponding angles are congruent). Both rigid motions and dilations preserve angle measures.

17. What does it mean if one pair of corresponding angles is not congruent?

    The triangles are not similar. All corresponding angles must be congruent for similarity.

18. What is the similarity ratio between areas of similar figures with scale factor $k$?

    $\frac{A_2}{A_1}=k^2$. Areas scale by the square of the side scale factor.

19. What is the side proportion statement for $\triangle ABC\sim\triangle DEF$?

    $\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$. Each ratio equals the common scale factor $k$.

20. What is the effect of a dilation on angles?

    Angles are preserved (remain congruent). Dilation only changes size, not angle measures.

21. Identify whether triangles with sides $3,4,5$ and $6,8,11$ are similar by proportionality.

    No; ratios are not all equal (for example, $\frac{11}{5}\ne\frac{6}{3}$). The ratios $\frac{11}{5}$ and $\frac{6}{3}$ are not equal.

22. Which transformations preserve distances exactly (not just proportionally)?

    Rigid motions: translations, rotations, and reflections. These transformations preserve all distances exactly.

23. What is the scale factor $k$ between similar figures?

    $k=\frac{\text{image length}}{\text{preimage length}}$ for any pair of corresponding sides. The ratio of any corresponding lengths gives the constant scale factor.

24. Identify the correspondence if $\triangle PQR\sim\triangle RST$ is written in that order.

    $P\leftrightarrow R$, $Q\leftrightarrow S$, $R\leftrightarrow T$. The order of vertices indicates the correspondence pattern.

25. Identify whether two triangles are similar if their corresponding angles are $40^\circ,60^\circ,80^\circ$ in both.

    Yes; all corresponding angles are congruent. Identical corresponding angles guarantee triangle similarity (AAA).

26. Identify whether $\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$ is sufficient to claim similarity for triangles.

    Yes; proportional corresponding sides imply similarity for triangles. For triangles, proportional sides guarantee similarity (SSS similarity).

27. Find $k$ if a preimage side is $12$ and the image side is $9$ under dilation.

    $k=\frac{9}{12}=\frac{3}{4}$. Scale factor is image divided by preimage for any dilation.

28. Identify the constant of proportionality in $\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}=k$.

    The scale factor $k$ from $\triangle ABC$ to $\triangle DEF$. This constant ratio defines the scale factor between similar triangles.

29. Find $A_2$ if similar figures have $A_1=20$ and side scale factor $k=\frac{3}{2}$.

    $A_2=20\left(\frac{3}{2}\right)^2=45$. Area scales by $k^2$: $20 \times (\frac{3}{2})^2 = 20 \times \frac{9}{4} = 45$.

30. Identify whether lengths $4,6,8$ and $6,9,12$ can be corresponding sides of similar triangles.

    Yes; all ratios equal $\frac{6}{4}=\frac{9}{6}=\frac{12}{8}=\frac{3}{2}$. All three ratios equal $\frac{3}{2}$, confirming proportionality.

31. What is the meaning of $\triangle ABC\sim\triangle DEF$ about correspondence?

    $A\leftrightarrow D$, $B\leftrightarrow E$, $C\leftrightarrow F$. The order of vertices indicates which angles and sides correspond.

32. What is the effect of a rigid motion on a figure's size and shape?

    Size and shape are unchanged; all distances and angles are preserved. Rigid motions create congruent figures with identical measurements.

33. What does a dilation with negative scale factor $k<0$ do to orientation?

    It reverses orientation and scales lengths by $|k|$. Negative scale factors flip the figure across the center of dilation.

34. Find $DE$ if $\triangle ABC\sim\triangle DEF$, $AB=8$, and $k=\frac{DE}{AB}=\frac{5}{4}$.

    $DE=10$. Multiply the preimage side by the scale factor: $8 \times \frac{5}{4} = 10$.

35. What transformations are allowed in a similarity transformation sequence?

    Rigid motions (translate, rotate, reflect) and dilations. These preserve shape and allow controlled size changes through dilation.

36. What is the effect of a dilation with $k>1$ on a figure?

    It enlarges the figure by factor $k$. Scale factor greater than 1 increases all dimensions.

37. What is the angle congruence statement for $\triangle ABC\sim\triangle DEF$?

    $\angle A\cong\angle D$, $\angle B\cong\angle E$, $\angle C\cong\angle F$. Corresponding angles are equal when triangles are similar.

38. Identify whether a translation followed by a rotation can change a figure's size.

    No; rigid motions preserve size. Rigid motions only change position and orientation, not size.

39. What is preserved by similarity transformations but not necessarily equal in measure?

    Lengths are preserved up to a common factor $k$ (proportional, not equal). Lengths are multiplied by a constant factor, maintaining proportionality.

40. Find the preimage length if the image length is $21$ under dilation with $k=3$.

    $7$. Divide the image length by the scale factor.

41. Find and correct the false claim: "Similar triangles must have equal perimeters."

    Correct: Perimeters are proportional; $\frac{P_2}{P_1}=k$. Similar triangles have proportional, not necessarily equal, perimeters.

42. Find and correct the false claim: "A dilation changes angle measures."

    Correct: A dilation preserves angle measures. Dilation preserves angles while changing side lengths proportionally.

43. Choose the correct ratio for $\triangle PQR\sim\triangle RST$ using the stated order.

    $\frac{PQ}{RS}=\frac{QR}{ST}=\frac{PR}{RT}$. Match corresponding sides based on vertex order.

44. What does it mean for two triangles to be similar in terms of angles?

    All corresponding angles are congruent. Angle measures are preserved under similarity transformations.

45. What property of lengths is preserved by a dilation with scale factor $k$?

    All lengths are multiplied by $k$. Dilation uniformly scales all distances by the same factor $k$.

46. What is the effect of a dilation with $0<k<1$ on a figure?

    It reduces the figure by factor $k$. Scale factor between 0 and 1 decreases all dimensions.

47. Identify the scale factor if corresponding sides are $14$ (image) and $-7$ (preimage direction ignored).

    $|k|=\frac{14}{7}=2$. Use absolute value since direction doesn't affect scale magnitude.

48. What must be true about corresponding angles if two figures are similar?

    Each pair of corresponding angles must be congruent. This is a requirement for similarity by definition.

49. What does it mean if corresponding side ratios are not equal for two triangles?

    The triangles are not similar. Unequal ratios violate the proportionality requirement for similarity.

50. Find the image perimeter if $P_1=36$ and the dilation scale factor is $k=\frac{5}{6}$.

    $P_2=36\cdot\frac{5}{6}=30$. Multiply the original perimeter by the scale factor.

51. Identify the missing ratio if $\triangle ABC\sim\triangle DEF$ and $\frac{AB}{DE}=\frac{2}{3}$.

    $\frac{BC}{EF}=\frac{2}{3}$. All corresponding side ratios must be equal in similar triangles.

52. Find the perimeter scale factor if corresponding side scale factor is $k=\frac{4}{5}$.

    $\frac{4}{5}$. Perimeters scale by the same factor as corresponding sides.

53. What is the definition of similarity using similarity 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.

54. What does a dilation with $k=1$ do to a figure?

    No change; it is the identity dilation. Scale factor of 1 produces a figure identical to the original.

55. Which option is a valid way to prove two figures are similar using transformations?

    Show a dilation and rigid motions map one figure exactly onto the other. Showing this mapping proves the figures are similar.

56. Choose the correct conclusion: $\angle A\cong\angle D$, $\angle B\cong\angle E$, $\angle C\cong\angle F$.

    $\triangle ABC\sim\triangle DEF$. All corresponding angles are congruent, so triangles are similar.

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