Determining Similarity via Transformations - Math 2

1. Two regular hexagons are positioned in the coordinate plane. Hexagon A has vertices at distance 6 from the center, while hexagon B has vertices at distance 4 from the center. Both hexagons are oriented so that one vertex lies on the positive x-axis. A student claims that a dilation centered at the origin with scale factor $\frac{2}{3}$ maps hexagon A onto hexagon B. How should this claim be evaluated?
   Answer: The claim is correct because the ratio of the circumradii is $\frac{4}{6} = \frac{2}{3}$, and both hexagons have the same orientation Regular hexagons are similar to each other, and the scale factor between them is the ratio of corresponding linear measurements. Since hexagon A has circumradius 6 and hexagon B has circumradius 4, the scale factor from A to B is 4/6 = 2/3. Both hexagons are oriented with a vertex on the positive x-axis and presumably centered at the origin, so a dilation by factor 2/3 centered at the origin will map A onto B. Choice B incorrectly suggests additional rotation is needed when orientations already match. Choice C confuses the direction of mapping. Choice D adds unnecessary conditions that are already met.
2. Two regular hexagons have side lengths of 4 cm and 10 cm respectively. A student claims they are similar because 'all regular hexagons have the same angles.' To prove similarity using transformations, what additional step is required beyond verifying equal angles?
   Answer: Demonstrate that a dilation by scale factor 2.5 followed by congruent transformations maps one hexagon onto the other To prove similarity via transformations, we must show that one figure can be mapped onto the other using similarity transformations. The scale factor is 10/4 = 2.5, so we need dilation by 2.5 followed by rigid motions (translations, rotations, reflections) to align the hexagons. Choice B involves area relationships but doesn't prove transformational similarity. Choice C verifies proportionality but not the transformation sequence. Choice D describes rotational symmetry, not similarity between different hexagons.
3. Circle A has radius 4 and circle B has radius 10. A similarity transformation maps circle A onto circle B. If the transformation includes a translation by vector $\langle 3, -2 \rangle$ and a rotation of 45° about point P, what additional transformation is required?
   Answer: A dilation by scale factor $\frac{5}{2}$ about any point, applied either before or after the other transformations To map circle A (radius 4) onto circle B (radius 10), we need a scale factor of 10/4 = 5/2. Since circles are preserved under rotation and translation (rigid motions), and these transformations don't change size, we need a dilation by factor 5/2. The order doesn't matter for achieving similarity - dilation can be applied before or after the rigid motions. The center of dilation can be any point since we have translation available to adjust position. Choice A uses incorrect scale factor. Choice C unnecessarily restricts the order and center. Choice D adds an unnecessary reflection.
4. Triangle JKL has sides of length 6, 8, and 10. Triangle MNP has sides of length 9, 12, and 15. A student claims these triangles are similar because there exists a dilation that maps one to the other. Which statement best evaluates this claim?
   Answer: The claim is correct; a dilation by scale factor $\frac{3}{2}$ maps triangle JKL to triangle MNP, proving similarity First, check if sides are proportional: 9/6 = 3/2, 12/8 = 3/2, 15/10 = 3/2. The sides are proportional with ratio 3:2, so a dilation by factor 3/2 maps triangle JKL to triangle MNP. Both triangles are right triangles (6²+8²=10² and 9²+12²=15²), so they have the same shape. Since corresponding sides are proportional, the triangles are similar. Choice A uses wrong scale factor direction. Choice C incorrectly suggests different shapes. Choice D incorrectly states that dilation alone cannot establish similarity when the triangles are already in the same orientation.
5. Circle O with radius 6 is transformed by a dilation with scale factor $\frac{3}{4}$ followed by a translation. The resulting circle O' has what relationship to the original circle O in terms of similarity?
   Answer: The circles are similar with O' having radius $\frac{9}{2}$ but a different center than O After dilation by 3/4, the radius becomes 6 × (3/4) = 9/2. Translation moves the center but doesn't affect the radius. All circles are similar to each other regardless of size, so the circles are similar. The center changes due to translation. Choice A incorrectly states the center remains the same. Choice C incorrectly calls them congruent when they have different radii. Choice D incorrectly suggests translation direction affects similarity.
6. Quadrilateral PQRS undergoes a dilation with scale factor k followed by a rotation. The resulting figure is quadrilateral P'Q'R'S'. If PQ = 8, QR = 12, and P'Q' = 6, what must be true for the quadrilaterals to be similar?
   Answer: Q'R' = 9 and the angles of PQRS must equal the corresponding angles of P'Q'R'S' Since P'Q' = 6 and PQ = 8, the scale factor k = 6/8 = 3/4. Therefore Q'R' = (3/4)(12) = 9. Similarity transformations preserve angles, so corresponding angles must be equal. The figures are already similar due to the transformation sequence. Choice B is redundant since proportional sides are guaranteed by the dilation. Choice C has wrong side length and perimeter condition is irrelevant. Choice D has wrong side length.
7. Two rhombuses have the same angles but different side lengths. The first rhombus has side length s, and the second has side length 3s. A student claims they can prove similarity by applying a dilation with scale factor 3. What additional information or step is needed to complete the similarity proof?
   Answer: Confirm that a sequence of rigid motions can map the dilated first rhombus onto the second After dilation by scale factor 3, the first rhombus will have side length 3s, matching the second rhombus. However, they may have different positions and orientations. To complete the similarity proof via transformations, we must show that rigid motions (translation, rotation, reflection) can align the dilated figure with the second rhombus. Choice A describes a property of all rhombuses, not a similarity requirement. Choice C verifies a consequence of similarity but doesn't establish it. Choice D is unnecessary since dilation preserves angles.
8. A regular pentagon with side length 12 undergoes a similarity transformation to produce a regular pentagon with side length 8. If the transformation consists of a dilation followed by a reflection across line ℓ, what was the scale factor of the dilation?
   Answer: $\frac{2}{3}$, because this is the ratio of corresponding sides after the complete transformation When you encounter similarity transformations, remember that the overall scale factor depends only on transformations that change size—reflections, rotations, and translations preserve all distances and don't affect scaling. In this problem, you need to find what scale factor was used in the dilation. Since the complete transformation (dilation followed by reflection) takes a pentagon with side length 12 to one with side length 8, the overall scale factor is $\frac{8}{12} = \frac{2}{3}$. Because reflections preserve all distances—they only flip orientation—the reflection doesn't change any lengths. Therefore, the dilation alone must account for the entire size change from 12 to 8, making the scale factor $\frac{2}{3}$. Choice A incorrectly assumes reflections reduce distances, which is false—reflections are rigid transformations that preserve all measurements. Choice B makes the same error, thinking the dilation must "compensate" for some reduction caused by reflection. Choice C presents $\frac{4}{6}$ (which equals $\frac{2}{3}$) but incorrectly suggests that reflection changes the "effective scale factor," implying reflections somehow modify the dilation's effect. Choice D correctly recognizes that $\frac{2}{3}$ is simply the ratio of final to initial side lengths, and since only the dilation changes size, this must be the dilation's scale factor. Study tip: Remember that in any sequence of transformations, only dilations affect the scale factor. Reflections, rotations, and translations are rigid transformations that preserve distances, so they don't contribute to overall size changes.
9. Quadrilateral PQRS undergoes a sequence of transformations to produce quadrilateral P'Q'R'S'. The transformations are: (1) dilation by scale factor 3 centered at point T, (2) rotation 90° clockwise about point U, (3) reflection across line m. If PQRS and P'Q'R'S' have the same orientation, what can be concluded about the relationship between these quadrilaterals?
   Answer: They are similar with scale factor 3, and the combination of rotation and reflection resulted in a net rotation The dilation by scale factor 3 makes P'Q'R'S' similar to PQRS with ratio 3:1. The rotation (90° clockwise) changes orientation, but the subsequent reflection also changes orientation. Two orientation-reversing transformations combine to preserve orientation overall. Since the final quadrilaterals have the same orientation as stated, the net effect of the rotation and reflection must be orientation-preserving (equivalent to a rotation). The quadrilaterals are similar with scale factor 3. Choice A is wrong because dilation changes size. Choice B incorrectly states the combination results in reflection. Choice D is wrong because rotation followed by reflection can preserve orientation.
10. Parallelogram JKLM undergoes a dilation with center P and scale factor 0.8, followed by a rotation of 45° about point Q. The resulting figure is parallelogram J'K'L'M'. If JK = 10 and the transformation sequence maps JKLM onto J'K'L'M', what is the length of J'K'?
    Answer: 8, because only the dilation affects side lengths and rotation preserves distances When you encounter transformation sequences, the key is understanding how each transformation affects the figure's properties. Some transformations preserve distances, while others change them. A dilation with scale factor 0.8 multiplies every length in the figure by 0.8. Since JK = 10, after dilation the corresponding side length becomes 10 × 0.8 = 8. The rotation that follows is an isometry (rigid transformation), meaning it preserves all distances and angles—it simply changes the figure's orientation without affecting any measurements. Therefore, J'K' = 8, making answer C correct. Only the dilation affects side lengths, while rotation preserves distances. Answer A incorrectly assumes rotation changes lengths by a factor of $\sqrt{2}$. This confuses rotation with some other transformation—rotations never change distances, regardless of the angle. Answer B suggests the two transformations somehow cancel each other out to restore the original size. This reflects a fundamental misunderstanding: transformations don't "undo" each other unless specifically designed as inverses, which isn't the case here. Answer D implies that rotation affects the scale factor of the dilation. This shows confusion about the order of operations—each transformation applies to the result of the previous one, and rotation cannot retroactively change how the dilation affected the figure. Study tip: Remember that rigid transformations (rotations, reflections, translations) always preserve distances and angles, while non-rigid transformations like dilations change sizes. When transformations are performed in sequence, apply them one at a time in the given order.
11. Rectangle ABCD has dimensions 6 × 9. Rectangle EFGH has dimensions 4 × 6. A student applies a dilation with scale factor $\frac{2}{3}$ to rectangle ABCD, creating rectangle A'B'C'D'. Which statement correctly describes the relationship between A'B'C'D' and EFGH?
    Answer: The rectangles are congruent because both have dimensions 4 × 6 after the dilation is applied After dilation by 2/3, rectangle A'B'C'D' has dimensions (2/3)(6) × (2/3)(9) = 4 × 6, which exactly matches rectangle EFGH. Since they have identical dimensions, they are congruent (and therefore also similar). Choice B incorrectly suggests they're only similar. Choice C incorrectly states dilation changes aspect ratio (it preserves ratios). Choice D suggests additional work is needed when the relationship is already clear.
12. Triangle ABC has vertices A(1,2), B(4,2), C(2.5,5). Triangle DEF has vertices D(0,0), E(6,0), F(3,6). To prove these triangles are similar using transformations, which approach would be most efficient?
    Answer: Dilation by scale factor 2 at origin, then rigid motions for alignment When proving triangle similarity using transformations, you need to identify the most direct sequence that maps one triangle onto the other through a combination of similarity transformations (dilations) and rigid motions (translations, rotations, reflections). Looking at the coordinates, triangle ABC has side lengths of 3, 3.5, and 3.5 units, while triangle DEF has side lengths of 6, 7, and 7 units. This immediately reveals a scale factor of 2 between corresponding sides. Since triangle DEF is already positioned with vertex D at the origin and one side along the x-axis, starting with a dilation by factor 2 at the origin is the most efficient approach. After this dilation, you'd only need rigid motions to complete the alignment. Choice A uses distance formulas and angle verification, which proves similarity but doesn't establish the transformation sequence efficiently. Choice B separates the similarity proof from finding transformations, creating unnecessary extra work. Choice C suggests translating A to the origin first, but this ignores that triangle DEF is already conveniently positioned at the origin, making this approach less efficient. Choice D correctly recognizes that dilation by scale factor 2 at the origin takes advantage of DEF's position and the clear 2:1 ratio between corresponding sides. After this single dilation, only simple rigid motions are needed to complete the mapping. Strategy tip: When comparing triangles for similarity transformations, first check if one triangle is positioned at the origin and calculate side ratios immediately. This often reveals the most efficient transformation sequence.
13. Triangle XYZ has sides of length 9, 12, and 15. Triangle MNP has sides of length 6, 8, and 10. After applying a dilation centered at point O with scale factor $\frac{2}{3}$, triangle XYZ becomes triangle X'Y'Z'. What transformation would map X'Y'Z' onto triangle MNP?
    Answer: A sequence of rigid motions (rotation, reflection, and/or translation) since the triangles are now congruent After dilation by 2/3, triangle X'Y'Z' has sides 6, 8, and 10, which exactly match triangle MNP. Since the triangles are now congruent (same side lengths), only rigid motions are needed to map one onto the other. Choice A ignores potential orientation differences. Choice B incorrectly suggests no transformation exists. Choice D applies unnecessary additional dilation when the triangles are already congruent.

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