Middle School Math Quiz: Understand Similarity Through Transformations
20 questions · exam conditions
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Understand Similarity Through TransformationsQuestion 1 of 20

Rectangle PQRS is similar to rectangle WXYZ. If the ratio of corresponding sides is 3:5, and rectangle PQRS can be mapped onto rectangle WXYZ through a sequence of transformations, which transformation must be included in this sequence?

A translation by vector (2, -3) to align the rectangles properly after scaling
A dilation with scale factor 53\frac{5}{3} to adjust the size difference between rectangles
A reflection across the line y = x to change the orientation of the rectangle
A rotation of 90° counterclockwise to match the corresponding vertex positions
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Middle School Math Quiz

Middle School Math Quiz: Understand Similarity Through Transformations

Practice Understand Similarity Through Transformations in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Understand Similarity Through Transformations, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Rectangle PQRS is similar to rectangle WXYZ. If the ratio of corresponding sides is 3:5, and rectangle PQRS can be mapped onto rectangle WXYZ through a sequence of transformations, which transformation must be included in this sequence?

  1. A translation by vector (2, -3) to align the rectangles properly after scaling
  2. A dilation with scale factor 53\frac{5}{3} to adjust the size difference between rectangles (correct answer)
  3. A reflection across the line y = x to change the orientation of the rectangle
  4. A rotation of 90° counterclockwise to match the corresponding vertex positions
Explanation: Since the rectangles are similar with a ratio of corresponding sides of 3:5, rectangle PQRS must be dilated by scale factor 5/3 to match the size of rectangle WXYZ. This dilation is essential for similarity transformations. Choice A describes a specific translation that may or may not be needed depending on position. Choice C describes a reflection that may or may not be needed depending on orientation. Choice D describes a rotation that may or may not be needed depending on orientation. Only the dilation is guaranteed to be required.

Question 2

Triangle ABC is similar to triangle A'B'C'. Point A is at (4, 6), and point A' is at (-2, -3). If the transformation sequence includes only a dilation centered at the origin followed by a reflection, what is the scale factor of the dilation?

  1. The scale factor is 12\frac{1}{2} because the distance from origin to A' is half the distance to A (correct answer)
  2. The scale factor is 12\frac{1}{2} because we must account for proportional coordinate changes
  3. The scale factor is 12\frac{1}{2} because both coordinates of A' have half the absolute value of A
  4. The scale factor cannot be determined without knowing the reflection line used
Explanation: To find the scale factor, we compare distances from the origin. Point A is at distance 42+62=52=213\sqrt{4^2 + 6^2} = \sqrt{52} = 2\sqrt{13} from origin. Point A' is at distance (−2)2+(−3)2=13\sqrt{(-2)^2 + (-3)^2} = \sqrt{13} from origin. The scale factor is 13213=12\frac{\sqrt{13}}{2\sqrt{13}} = \frac{1}{2}. Choice A correctly identifies that scale factor is determined by the ratio of distances from the center of dilation.

Question 3

Triangle JKLJKL has side lengths 44, 66, and 88. Triangle MNOMNO has side lengths 66, 99, and 1212. Are the triangles similar? If yes, what is the scale factor from △JKL\triangle JKL to △MNO\triangle MNO?

  1. No; the triangles are not similar because translations are required.
  2. No; the triangles are not similar because 4+6≠84+6\ne 8.
  3. Yes; scale factor k=32k=\tfrac{3}{2}. (correct answer)
  4. Yes; scale factor k=23k=\tfrac{2}{3}.
Explanation: This question tests understanding of similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures have the same shape with proportional sides (corresponding sides have equal ratios forming scale factor k) and equal corresponding angles; it requires dilation since rigid transformations (rotation, reflection, translation) preserve size giving congruence (k=1), while dilation scales by factor k creating different sizes (k=2 doubles all lengths, k=1/2 halves); the sequence is typically 'dilate by k from center, then rotate/reflect/translate as needed to position' or variations—dilation creates size difference, others adjust position/orientation. For example, a triangle with sides 3-4-5 is similar to a triangle with sides 6-8-10; check proportionality: 6/3=8/4=10/5=2 (equal ratios, scale factor k=2), sequence could be 'dilate by 2 from origin' giving similar triangle 2× larger, then translate/rotate to match position if needed. The triangles are similar with scale factor 3/2 from JKL to MNO since 6/4=3/2, 9/6=3/2, 12/8=3/2, making choice A correct. A common error is using sum instead of ratio (like 4+6≠8) or thinking translations prevent similarity. To check similarity: (1) measure corresponding sides (assume 4 to 6, 6 to 9, 8 to 12), (2) calculate ratios (6/4=1.5, etc.), (3) verify equal (yes, k=3/2), (4) angles equal by proportionality. Mistakes include inverting k to 2/3 or claiming not similar due to translations.

Question 4

Triangle ABCABC has side lengths AB=3AB=3, BC=4BC=4, and CA=5CA=5. Triangle DEFDEF has side lengths DE=6DE=6, EF=8EF=8, and FD=10FD=10. Which statement is true?

  1. The triangles are not similar because a dilation changes the angles.
  2. The triangles are congruent because all corresponding sides are different.
  3. The triangles are similar; △DEF\triangle DEF is a dilation of △ABC\triangle ABC with scale factor 22. (correct answer)
  4. The triangles are similar; △DEF\triangle DEF is a dilation of △ABC\triangle ABC with scale factor 12\tfrac{1}{2}.
Explanation: This question tests understanding of similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures have the same shape with proportional sides (corresponding sides have equal ratios forming scale factor k) and equal corresponding angles; it requires dilation since rigid transformations (rotation, reflection, translation) preserve size giving congruence (k=1), while dilation scales by factor k creating different sizes (k=2 doubles all lengths, k=1/2 halves); the sequence is typically 'dilate by k from center, then rotate/reflect/translate as needed to position' or variations—dilation creates size difference, others adjust position/orientation. For example, a triangle with sides 3-4-5 is similar to a triangle with sides 6-8-10; check proportionality: 6/3=8/4=10/5=2 (equal ratios, scale factor k=2), sequence could be 'dilate by 2 from origin' giving similar triangle 2× larger, then translate/rotate to match position if needed. Here, the triangles are similar with scale factor 2 since 6/3=2, 8/4=2, 10/5=2, so triangle DEF is a dilation of triangle ABC with k=2, making choice B correct. A common error is confusing congruence and similarity, like claiming they are congruent despite different sizes or thinking dilation changes angles, which it doesn't. To check similarity: (1) measure corresponding sides (AB to DE, BC to EF, CA to FD), (2) calculate ratios (3/6=1/2, but actually for mapping ABC to DEF it's 6/3=2), (3) verify equal (all ratios 2, yes—similar with k=2), (4) angles are equal since both are right triangles (3-4-5 is right-angled). Transformation sequence: identify scale factor (k=6/3=2), describe dilation (dilate by 2 from a center), add rigid transformations if needed; congruence is similarity with k=1, but here k=2 so similar but not congruent; mistakes include inverting scale factor or claiming not similar due to angle change.

Question 5

Triangle XYZXYZ has vertices X(2,0)X(2,0), Y(4,0)Y(4,0), and Z(2,3)Z(2,3). A dilation with scale factor 12\dfrac{1}{2} about the origin is applied to triangle XYZXYZ to create triangle X′Y′Z′X'Y'Z'. What are the coordinates of Y′Y'?

  1. (2,0)(2,0) (correct answer)
  2. (1,0)(1,0)
  3. (4,0)(4,0)
  4. (2,32)(2,\dfrac{3}{2})
Explanation: Tests understanding similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures: same shape, proportional sides (corresponding sides have equal ratios forming scale factor k), equal corresponding angles. Requires dilation: rigid transformations (rotation, reflection, translation) preserve size giving congruence (k=1), dilation scales by factor k creating different sizes (k=2 doubles all lengths, k=1/2 halves). Sequence: typically "dilate by k from center, then rotate/reflect/translate as needed to position" or variations—dilation creates size difference, others adjust position/orientation. [Example: triangle with sides 3-4-5 similar to triangle with sides 6-8-10, check proportionality: 6/3=8/4=10/5=2 (equal ratios, scale factor k=2), sequence could be "dilate by 2 from origin" giving similar triangle 2× larger, then translate/rotate to match position if needed]. The coordinates of Y' after dilation by 1/2 about the origin are (4/2, 0/2) = (2,0). A common error is misapplying the dilation formula, like halving only one coordinate or confusing with other transformations. Checking similarity: (1) measure corresponding sides (side AB corresponds to A'B', BC to B'C', etc.), (2) calculate ratios (AB/A'B', BC/B'C', CA/C'A'), (3) verify equal (all ratios same value k? yes→similar with scale factor k), (4) check angles if uncertain (corresponding angles equal? yes→similar). Transformation sequence: identify scale factor (ratio of sides: k=6/3=2), describe dilation (dilate by 2 from origin), add rigid transformations if needed (rotate, reflect, translate to match position). Congruence is similarity with k=1 (special case: same size and shape). Mistakes: forgetting dilation (trying to use only rigid for different sizes—impossible), inverting scale factor (using smaller/larger instead of larger/smaller), claiming proportional when ratios differ (not checking all pairs).

Question 6

Circle P has radius 4 and center at (3, 5). Circle Q has radius 6 and center at (-2, -1). A student claims these circles are similar and that circle P can be mapped onto circle Q. Which sequence of transformations would accomplish this mapping?

  1. Translation by vector (-5, -6), then dilation by scale factor 1.5 centered at the new position (correct answer)
  2. Dilation by scale factor 1.5 centered at (3, 5), then translation by vector (-5, -6)
  3. Dilation by scale factor 1.5 centered at the origin, then translation by vector (-6.5, -8.5)
  4. Translation by vector (-5, -6), then dilation by scale factor 0.67 centered at the origin
Explanation: To map circle P onto circle Q: First, translate P's center from (3, 5) to Q's center (-2, -1) using vector (-5, -6). Then dilate by scale factor 6/4 = 1.5 to match the radius. When dilating after translation, the center of dilation should be at the current position of the circle's center. Choice B dilates first, which moves the center away from the desired final position. Choice C uses the wrong translation vector. Choice D uses the wrong scale factor (0.67 instead of 1.5).

Question 7

Triangle ABCABC has vertices A(1,1)A(1,1), B(4,1)B(4,1), and C(1,3)C(1,3). Triangle A′B′C′A'B'C' has vertices A′(2,2)A'(2,2), B′(8,2)B'(8,2), and C′(2,6)C'(2,6). Which sequence of transformations maps triangle ABCABC to triangle A′B′C′A'B'C'?

  1. Dilate by scale factor 22 about the origin, then translate right 0 and up 0 (correct answer)
  2. Translate right 1 and up 1, then rotate 90∘90^\circ counterclockwise about the origin
  3. Translate right 1 and up 1, then dilate by scale factor 22 about the origin
  4. Reflect across the yy-axis, then dilate by scale factor 22 about the origin
Explanation: Checking whether dilating by scale factor 2 about the origin maps each vertex of triangle ABCABC to triangle A′B′C′A'B'C': A(1,1)→(2,2)=A′A(1,1)\to(2,2)=A', B(4,1)→(8,2)=B′B(4,1)\to(8,2)=B', C(1,3)→(2,6)=C′C(1,3)\to(2,6)=C'. Every vertex matches with no translation needed, so the correct sequence is a dilation by scale factor 2 about the origin alone. Choice B is incorrect because it adds an unnecessary rotation. Choice C is incorrect because translating before dilating changes the result: translating A(1,1)A(1,1) to (2,2)(2,2) first and then dilating by 2 about the origin gives (4,4)(4,4), not A′(2,2)A'(2,2). Choice D is incorrect because reflecting across the yy-axis before dilating sends A(1,1)A(1,1) to (−2,2)(-2,2), not A′(2,2)A'(2,2).

Question 8

Two similar parallelograms are shown on the coordinate grid. The first parallelogram has vertices at A(2, 1), B(5, 1), C(6, 3), and D(3, 3). To map this parallelogram onto the second one, which combination of transformations is needed?

  1. Dilation by scale factor 2 centered at origin, then reflection across line y = -x
  2. Reflection across the x-axis, then dilation by scale factor 2 centered at origin
  3. Dilation by scale factor 2 centered at origin, then rotation of 180° about origin (correct answer)
  4. Rotation of 180° about origin, then dilation by scale factor 2 centered at origin
Explanation: To map the first parallelogram onto the second: First, dilate by scale factor 2 centered at the origin to double the size. This maps A(2,1)→(4,2), B(5,1)→(10,2), C(6,3)→(12,6), D(3,3)→(6,6). Then rotate 180° about the origin to achieve the correct orientation: (4,2)→(-4,-2), (10,2)→(-10,-2), (12,6)→(-12,-6), (6,6)→(-6,-6). This matches the second parallelogram's position in quadrant III.

Question 9

Two triangles are related by a dilation with scale factor k=1k=1. Which statement best describes the relationship between the triangles?

  1. They are neither similar nor congruent.
  2. They are similar but not congruent because dilations always change size.
  3. They are congruent and therefore similar. (correct answer)
  4. They must be reflections of each other rather than images under a dilation.
Explanation: This question tests understanding of similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures have the same shape with proportional sides (corresponding sides have equal ratios forming scale factor k) and equal corresponding angles; it requires dilation since rigid transformations (rotation, reflection, translation) preserve size giving congruence (k=1), while dilation scales by factor k creating different sizes (k=2 doubles all lengths, k=1/2 halves); the sequence is typically 'dilate by k from center, then rotate/reflect/translate as needed to position' or variations—dilation creates size difference, others adjust position/orientation. For example, a triangle with sides 3-4-5 is similar to a triangle with sides 6-8-10; check proportionality: 6/3=8/4=10/5=2 (equal ratios, scale factor k=2), sequence could be 'dilate by 2 from origin' giving similar triangle 2× larger, then translate/rotate to match position if needed. With k=1, the triangles are congruent (same size and shape), and thus also similar, making choice B correct. A common error is confusing congruence and similarity, like claiming they are similar but not congruent when k=1 means they are both. Congruence is similarity with k=1 (special case: same size and shape). Mistakes include thinking dilation with k=1 changes something or that it implies reflection only.

Question 10

Two similar pentagons have a scale factor of 2:3. If the first pentagon undergoes a sequence of transformations to map onto the second pentagon, and the sequence includes a dilation, a rotation of 45°, and a translation, in which order must these transformations be applied?

  1. The order does not matter because all transformations preserve similarity
  2. Dilation first, then rotation and translation in any order
  3. Translation last, with dilation and rotation in either order first (correct answer)
  4. Dilation last because it must follow positioning transformations
Explanation: In similarity transformations, rigid transformations (rotations, reflections, translations) preserve shape and size, while dilations change size but preserve shape. Dilation and rotation can be applied in either order since both preserve the figure's center relative to the dilation point. However, translation typically comes last because it moves the entire figure to its final position after size and orientation adjustments are complete.

Question 11

Triangle XYZ undergoes a sequence of transformations to produce similar triangle X'Y'Z'. The sequence includes a reflection across the x-axis, a dilation by scale factor 34\frac{3}{4}, and a rotation of 60° counterclockwise. If triangle XYZ has an area of 48 square units, what is the area of triangle X'Y'Z'?

  1. The area of triangle X'Y'Z' is 12 square units because the scale factor reduces all measurements proportionally
  2. The area of triangle X'Y'Z' is 36 square units because area scales linearly with the dilation factor
  3. The area of triangle X'Y'Z' is 48 square units because reflections and rotations preserve area measurements
  4. The area of triangle X'Y'Z' is 27 square units because area scales by the square of the scale factor (correct answer)
Explanation: When you encounter transformation problems involving area, the key insight is understanding which transformations affect area and by how much. Different transformations have different effects on measurements. Let's work through this step-by-step. The triangle undergoes three transformations: reflection across the x-axis, dilation by scale factor 34\frac{3}{4}, and a 60° rotation. Here's what each does to area:
  • Reflection: Preserves all measurements, including area
  • Rotation: Preserves all measurements, including area
  • Dilation: Changes area by the square of the scale factor
Since only the dilation affects area, we calculate: Area of X'Y'Z' = Original area × (scale factor)² = 48 × (34)2\left(\frac{3}{4}\right)^2 = 48 × 916\frac{9}{16} = 27 square units. Looking at the wrong answers: Choice A incorrectly applies the scale factor directly to area (48 × 34\frac{3}{4} = 36, then somehow gets 12). Choice B makes the common error of thinking area scales linearly with the dilation factor, giving 48 × 34\frac{3}{4} = 36. Choice C ignores the dilation entirely, focusing only on the fact that reflections and rotations preserve area. Study tip: Remember that area always scales by the square of the linear scale factor in dilations. If a shape is dilated by factor kk, its area changes by factor k2k^2. Reflections and rotations never change area, but dilations always do unless the scale factor is 1.

Question 12

Triangle PQRPQR has side lengths PQ=6PQ=6, QR=9QR=9, and RP=12RP=12. Triangle P′Q′R′P'Q'R' has side lengths P′Q′=2P'Q'=2, Q′R′=3Q'R'=3, and R′P′=4R'P'=4. What is the scale factor that maps △PQR\triangle PQR to △P′Q′R′\triangle P'Q'R'?

  1. k=23k=\tfrac{2}{3}
  2. k=12k=\tfrac{1}{2}
  3. k=13k=\tfrac{1}{3} (correct answer)
  4. k=3k=3
Explanation: This question tests understanding of similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures have the same shape with proportional sides (corresponding sides have equal ratios forming scale factor k) and equal corresponding angles; it requires dilation since rigid transformations (rotation, reflection, translation) preserve size giving congruence (k=1), while dilation scales by factor k creating different sizes (k=2 doubles all lengths, k=1/2 halves); the sequence is typically 'dilate by k from center, then rotate/reflect/translate as needed to position' or variations—dilation creates size difference, others adjust position/orientation. For example, a triangle with sides 3-4-5 is similar to a triangle with sides 6-8-10; check proportionality: 6/3=8/4=10/5=2 (equal ratios, scale factor k=2), sequence could be 'dilate by 2 from origin' giving similar triangle 2× larger, then translate/rotate to match position if needed. The scale factor mapping PQR to P'Q'R' is 1/3 since 2/6=1/3, 3/9=1/3, 4/12=1/3, making choice A correct. A common error is inverting the scale factor, like using 3 instead of 1/3 by swapping which maps to which. To check similarity: (1) measure corresponding sides (PQ to P'Q', etc.), (2) calculate ratios (6/2=3, but for mapping PQR to P'Q'R' it's 2/6=1/3), (3) verify equal (all 1/3, yes—similar with k=1/3), (4) angles equal implicitly by proportionality. Mistakes include computing as difference (e.g., 6-2=4) instead of ratio or claiming not proportional when they are.

Question 13

On a coordinate plane, triangle ABCABC has points A(1,1)A(1,1), B(3,1)B(3,1), and C(2,3)C(2,3). Triangle A′B′C′A'B'C' has points A′(−2,−2)A'(-2,-2), B′(−6,−2)B'(-6,-2), and C′(−4,−6)C'(-4,-6). Which sequence of transformations maps △ABC\triangle ABC to △A′B′C′\triangle A'B'C'?

  1. Dilate by scale factor 12\tfrac{1}{2} about the origin, then translate by (−4,−4)(-4,-4).
  2. Reflect across the yy-axis, then translate by (−3,−3)(-3,-3).
  3. Translate by (−3,−3)(-3,-3), then rotate 90∘90^\circ clockwise about the origin.
  4. Dilate by scale factor 22 about the origin, then translate by (−4,−4)(-4,-4). (correct answer)
Explanation: This question tests understanding of similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures have the same shape with proportional sides (corresponding sides have equal ratios forming scale factor k) and equal corresponding angles; it requires dilation since rigid transformations (rotation, reflection, translation) preserve size giving congruence (k=1), while dilation scales by factor k creating different sizes (k=2 doubles all lengths, k=1/2 halves); the sequence is typically 'dilate by k from center, then rotate/reflect/translate as needed to position' or variations—dilation creates size difference, others adjust position/orientation. For example, a triangle with sides 3-4-5 is similar to a triangle with sides 6-8-10; check proportionality: 6/3=8/4=10/5=2 (equal ratios, scale factor k=2), sequence could be 'dilate by 2 from origin' giving similar triangle 2× larger, then translate/rotate to match position if needed. The correct sequence is dilate by scale factor 2 about the origin (e.g., A(1,1) to (2,2), but then translate by (-4,-4) to (-2,-2); similarly for others), making choice B correct. A common error is using only rigid transformations for different sizes, which is impossible, or wrong scale factor like 1/2. To check similarity: (1) measure corresponding sides (e.g., AB distance sqrt((3-1)^2+(1-1)^2)=2, A'B' sqrt((-6+2)^2+(-2+2)^2)=4, ratio 4/2=2), (2) calculate ratios (all 2), (3) verify equal (yes, k=2), (4) angles equal as shapes match. Transformation sequence: identify k=2, dilate by 2 from origin, then translate by (-4,-4); mistakes include forgetting dilation or inverting k.

Question 14

Triangle PQRPQR has vertices P(1,0)P(1,0), Q(3,0)Q(3,0), and R(1,2)R(1,2). Triangle P′Q′R′P'Q'R' has vertices P′(−2,1)P'(-2,1), Q′(4,1)Q'(4,1), and R′(−2,7)R'(-2,7). Which sequence of transformations maps △PQR\triangle PQR to △P′Q′R′\triangle P'Q'R'?

  1. Rotate 180∘180^\circ about the origin, then translate left 3 and up 1
  2. Dilate by scale factor 33 about the origin, then translate left 5 and up 1 (correct answer)
  3. Translate left 5 and up 1 only (no dilation needed)
  4. Translate left 3 and up 1, then dilate by scale factor 33 about the origin
Explanation: This question tests understanding of similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures have the same shape with proportional sides where corresponding sides have equal ratios forming scale factor k, and equal corresponding angles; it requires dilation because rigid transformations like rotation, reflection, and translation preserve size giving congruence with k=1, while dilation scales by factor k creating different sizes, such as k=2 doubling all lengths or k=1/2 halving them, with sequences typically involving dilate by k from a center then rotate, reflect, or translate as needed to position, where dilation creates the size difference and others adjust position or orientation. For example, a triangle with sides 3-4-5 is similar to one with sides 6-8-10, checking proportionality: 6/3=8/4=10/5=2 for equal ratios with scale factor k=2, and a sequence could be dilate by 2 from the origin giving a similar triangle twice as large, then translate or rotate to match position if needed. Here, dilating by scale factor 3 about the origin followed by translating left 5 and up 1 maps PQR to P'Q'R', as dilation takes (1,0) to (3,0) then to (-2,1), and similarly for others, so choice B is correct. A common error is misordering transformations like translating first in A, which scales the translation incorrectly, or using only rigid transformations without dilation in D, impossible for size change. To find the sequence, identify scale factor from side ratios like base 2 to 6 giving k=3, describe dilation by 3 from origin, then add translation by comparing dilated points to targets. Mistakes include forgetting dilation for different sizes or using wrong translation vectors.

Question 15

Rectangle RR is 2 cm2\text{ cm} by 4 cm4\text{ cm}. Rectangle SS is 3 cm3\text{ cm} by 5 cm5\text{ cm}. Are the rectangles similar?

  1. Yes, because you can translate and rotate one rectangle to match the other.
  2. No, because the side ratios are not equal: 2/3≠4/52/3\ne 4/5. (correct answer)
  3. Yes, all rectangles are similar.
  4. No, because dilations change angle measures.
Explanation: Tests understanding similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures: same shape, proportional sides (corresponding sides have equal ratios forming scale factor k), equal corresponding angles. Requires dilation: rigid transformations (rotation, reflection, translation) preserve size giving congruence (k=1), dilation scales by factor k creating different sizes (k=2 doubles all lengths, k=1/2 halves). Sequence: typically "dilate by k from center, then rotate/reflect/translate as needed to position" or variations—dilation creates size difference, others adjust position/orientation. [Example: triangle with sides 3-4-5 similar to triangle with sides 6-8-10, check proportionality: 6/3=8/4=10/5=2 (equal ratios, scale factor k=2), sequence could be "dilate by 2 from origin" giving similar triangle 2× larger, then translate/rotate to match position if needed]. The rectangles are not similar because the side ratios are not equal: 2/3 ≠ 4/5 for corresponding sides. A common error is assuming all rectangles are similar or that rigid transformations alone can make them similar without checking proportionality. Checking similarity: (1) measure corresponding sides (side AB corresponds to A'B', BC to B'C', etc.), (2) calculate ratios (AB/A'B', BC/B'C', CA/C'A'), (3) verify equal (all ratios same value k? yes→similar with scale factor k), (4) check angles if uncertain (corresponding angles equal? yes→similar). Transformation sequence: identify scale factor (ratio of sides: k=6/3=2), describe dilation (dilate by 2 from origin), add rigid transformations if needed (rotate, reflect, translate to match position). Congruence is similarity with k=1 (special case: same size and shape). Mistakes: forgetting dilation (trying to use only rigid for different sizes—impossible), inverting scale factor (using smaller/larger instead of larger/smaller), claiming proportional when ratios differ (not checking all pairs).

Question 16

Triangle ABCABC has side lengths 3,5,63,5,6. Triangle DEFDEF has side lengths 3,4,53,4,5. Are the triangles similar?

  1. Yes, because you can always translate one triangle to match the other
  2. No, because dilation changes angles
  3. Yes, because each triangle has three sides
  4. No, because the ratios of corresponding sides are not all equal (correct answer)
Explanation: This question tests understanding of similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures have the same shape with proportional sides where corresponding sides have equal ratios forming scale factor k, and equal corresponding angles; it requires dilation because rigid transformations like rotation, reflection, and translation preserve size giving congruence with k=1, while dilation scales by factor k creating different sizes, such as k=2 doubling all lengths or k=1/2 halving them, with sequences typically involving dilate by k from a center then rotate, reflect, or translate as needed to position, where dilation creates the size difference and others adjust position or orientation. For example, a triangle with sides 3-4-5 is similar to one with sides 6-8-10, checking proportionality: 6/3=8/4=10/5=2 for equal ratios with scale factor k=2, and a sequence could be dilate by 2 from the origin giving a similar triangle twice as large, then translate or rotate to match position if needed. In this case, the triangles are not similar because ratios like 3/3=1, 5/4=1.25, 6/5=1.2 are not equal, so no consistent k, making choice C correct. Common errors include assuming all triangles are similar like in A or B, or wrongly stating dilation changes angles in D, but dilation preserves angles. To check, calculate ratios of corresponding sides assuming possible orders, verify if all equal, and if not, they are not similar. Mistakes include not checking all ratios or confusing similarity with having the same number of sides.

Question 17

A small right triangle has side lengths 33, 44, and 55 inches. A larger right triangle has side lengths 66, 88, and 1010 inches. Which statement is true?

  1. The triangles are similar with scale factor 22 (larger compared to smaller). (correct answer)
  2. The triangles are congruent because they are both right triangles.
  3. The triangles are not similar because dilation changes angle measures.
  4. The triangles are similar with scale factor 33 because 6−3=36-3=3.
Explanation: Tests understanding similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures: same shape, proportional sides (corresponding sides have equal ratios forming scale factor k), equal corresponding angles. Requires dilation: rigid transformations (rotation, reflection, translation) preserve size giving congruence (k=1), dilation scales by factor k creating different sizes (k=2 doubles all lengths, k=1/2 halves). Sequence: typically "dilate by k from center, then rotate/reflect/translate as needed to position" or variations—dilation creates size difference, others adjust position/orientation. [Example: triangle with sides 3-4-5 similar to triangle with sides 6-8-10, check proportionality: 6/3=8/4=10/5=2 (equal ratios, scale factor k=2), sequence could be "dilate by 2 from origin" giving similar triangle 2× larger, then translate/rotate to match position if needed]. The triangles are similar with scale factor 2, as sides 6/3=2, 8/4=2, 10/5=2, and both are right triangles so angles match. A common error is confusing similarity with congruence or miscalculating scale factor as a difference like 6-3=3 instead of the ratio. Checking similarity: (1) measure corresponding sides (side AB corresponds to A'B', BC to B'C', etc.), (2) calculate ratios (AB/A'B', BC/B'C', CA/C'A'), (3) verify equal (all ratios same value k? yes→similar with scale factor k), (4) check angles if uncertain (corresponding angles equal? yes→similar). Transformation sequence: identify scale factor (ratio of sides: k=6/3=2), describe dilation (dilate by 2 from origin), add rigid transformations if needed (rotate, reflect, translate to match position). Congruence is similarity with k=1 (special case: same size and shape). Mistakes: forgetting dilation (trying to use only rigid for different sizes—impossible), inverting scale factor (using smaller/larger instead of larger/smaller), claiming proportional when ratios differ (not checking all pairs).

Question 18

Which statement correctly describes the difference between congruent figures and similar figures?

  1. Congruent figures can be made using only translations, but similar figures require a rotation
  2. Congruent figures have equal corresponding angles, but similar figures do not
  3. Congruent figures are similar with scale factor k=1k=1, while similar figures may have k≠1k\ne 1 (correct answer)
  4. Similar figures must have the same size, but congruent figures can have different sizes
Explanation: This question tests understanding of similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures have the same shape with proportional sides where corresponding sides have equal ratios forming scale factor k, and equal corresponding angles; it requires dilation because rigid transformations like rotation, reflection, and translation preserve size giving congruence with k=1, while dilation scales by factor k creating different sizes, such as k=2 doubling all lengths or k=1/2 halving them, with sequences typically involving dilate by k from a center then rotate, reflect, or translate as needed to position, where dilation creates the size difference and others adjust position or orientation. For example, a triangle with sides 3-4-5 is similar to one with sides 6-8-10, checking proportionality: 6/3=8/4=10/5=2 for equal ratios with scale factor k=2, and a sequence could be dilate by 2 from the origin giving a similar triangle twice as large, then translate or rotate to match position if needed. The correct difference is that congruent figures are similar with k=1, while similar figures may have k≠1, so choice C is accurate. Common errors include reversing the definitions like in D, or misstatement about angles or transformations in A and B. To distinguish, note both have equal angles, but similarity allows proportional sides with k≠1 via dilation, while congruence requires k=1 with rigid motions only. Congruence is a special case of similarity where sizes are identical.

Question 19

Triangle XYZXYZ has side lengths 77, 99, and 1212. Triangle X′Y′Z′X'Y'Z' has corresponding side lengths 1414, 1818, and 2424. What is the scale factor from △XYZ\triangle XYZ to △X′Y′Z′\triangle X'Y'Z'?

  1. k=5k=5
  2. k=2k=2 (correct answer)
  3. k=12k=\frac{1}{2}
  4. k=7k=7
Explanation: This question tests understanding of similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures have the same shape with proportional sides where corresponding sides have equal ratios forming scale factor k, and equal corresponding angles; it requires dilation because rigid transformations like rotation, reflection, and translation preserve size giving congruence with k=1, while dilation scales by factor k creating different sizes, such as k=2 doubling all lengths or k=1/2 halving them, with sequences typically involving dilate by k from a center then rotate, reflect, or translate as needed to position, where dilation creates the size difference and others adjust position or orientation. For example, a triangle with sides 3-4-5 is similar to one with sides 6-8-10, checking proportionality: 6/3=8/4=10/5=2 for equal ratios with scale factor k=2, and a sequence could be dilate by 2 from the origin giving a similar triangle twice as large, then translate or rotate to match position if needed. Here, the scale factor from XYZ to X'Y'Z' is k=2 since 14/7=2, 18/9=2, and 24/12=2, confirming similarity, so choice B is correct. Common errors include computing wrong ratios like adding sides or picking unrelated numbers like 5 or 7 in C and D. To check, calculate each pair's ratio and ensure they equal k=2, verifying angles if needed by triangle properties. Congruence is similarity with k=1, and mistakes include inverting k to 1/2 or claiming non-proportional sides are similar.

Question 20

Polygon AA is transformed into polygon BB using only a rotation and a translation (no dilation). Which statement must be true?

  1. Polygon BB is similar to polygon AA but not congruent.
  2. Polygon BB is congruent to polygon AA. (correct answer)
  3. Polygon BB must be larger than polygon AA.
  4. Polygon BB has angles different from polygon AA.
Explanation: This question tests understanding of similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures have the same shape with proportional sides (corresponding sides have equal ratios forming scale factor k) and equal corresponding angles; it requires dilation since rigid transformations (rotation, reflection, translation) preserve size giving congruence (k=1), while dilation scales by factor k creating different sizes (k=2 doubles all lengths, k=1/2 halves); the sequence is typically 'dilate by k from center, then rotate/reflect/translate as needed to position' or variations—dilation creates size difference, others adjust position/orientation. For example, a triangle with sides 3-4-5 is similar to a triangle with sides 6-8-10; check proportionality: 6/3=8/4=10/5=2 (equal ratios, scale factor k=2), sequence could be 'dilate by 2 from origin' giving similar triangle 2× larger, then translate/rotate to match position if needed. Since only rigid transformations (rotation and translation) are used, no dilation means same size, so polygon B is congruent to A, making choice B correct. A common error is claiming similar but not congruent without dilation, but rigid transformations give congruence (k=1). Congruence is similarity with k=1 (special case: same size and shape). Mistakes include thinking no dilation means angles change or that B must be larger.