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.

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.

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, dilating triangle ABC by scale factor 2 about the origin maps A(1,1) to (2,2), B(4,1) to (8,2), and C(1,3) to (2,6), exactly matching A'B'C' with no additional translation needed, so choice C is correct. A common error is choosing a sequence without dilation or misordering transformations, like translating first then dilating in choice D, which would not produce the correct coordinates since dilation after translation scales the translation vector as well. To check similarity and transformations, identify corresponding points, calculate the scale factor from ratios like distance AB=3 to A'B'=6 giving k=6/3=2, verify it for all sides, and confirm angles are preserved; here, the sequence is dilation by 2 from the origin with no rigid transformations needed. Remember, congruence is similarity with k=1 as a special case of same size and shape, and mistakes include forgetting dilation for size changes or inverting the scale factor.

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 -2 about the origin (which includes a reflection) maps L(2,1) to (-4,-2), M(5,1) to (-10,-2), and N(2,4) to (-4,-8), exactly matching with no translation, so choice C is correct. A common error is using positive k without reflection or adding unnecessary translation like in A, or omitting dilation in D. To check, identify scale factor |k|=2 from side ratios and note the orientation flip indicating negative k, then verify coordinates match after transformation. Mistakes include forgetting the reflection aspect of negative k or trying rigid transformations only for size changes.

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.

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 rectangles are not similar because the ratios of corresponding sides 2/3 ≠ 4/5, so no consistent scale factor exists, making choice C correct. Common errors include assuming all rectangles are similar like in A, using addition instead of ratios like in B, or believing only squares can be similar rectangles in D, confusing the need for proportional sides. To check similarity, identify corresponding sides such as shorter to shorter and longer to longer, calculate ratios 2/3 and 4/5, verify they are not equal, and note angles are all 90° but proportionality fails. Congruence is similarity with k=1 as a special case of same size and shape, and mistakes include claiming proportional when ratios differ or forgetting to check all pairs.

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 triangles with sides 3-4-5 and 6-8-10 are similar with scale factor k=2 from smaller to larger since all ratios are 2 and angles are equal as both are right triangles, so choice A is correct. Common errors include confusing similarity with congruence like in B despite different sizes, denying similarity due to size difference in C, or computing scale factor as a difference like 6-3=3 in D instead of the ratio 6/3=2. To check similarity, measure corresponding sides assuming order 3 to 6, 4 to 8, 5 to 10, calculate ratios 6/3=2, 8/4=2, 10/5=2, verify they are equal for k=2, and confirm corresponding angles are equal if needed. Congruence is similarity with k=1 as a special case of same size and shape, and mistakes include claiming proportional when ratios differ or inverting the scale factor.

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.

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 3 about origin (small square side 1 becomes 3, e.g., (0,0) to (0,0), (1,0) to (3,0)), then translate by (5,5) to match (5,5), (8,5), etc., making choice B correct. A common error is switching order (translating first then dilating changes the translation vector due to scaling). To check similarity: (1) measure corresponding sides (small side 1, large side 3), (2) calculate ratios (3/1=3), (3) verify equal (yes, k=3), (4) angles equal (both squares). Transformation sequence: identify k=3, dilate by 3 from origin, then translate by (5,5); mistakes include using only rigid transformations for size change or wrong order.

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.