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).

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 (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.

Tests understanding similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor $k \neq 1$. 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 $2 \times$ larger, then translate/rotate to match position if needed]. The polygons are congruent since dilation by $k=1$ leaves size unchanged, and congruence is a special case of similarity. A common error is thinking dilation with $k=1$ changes size or that similarity requires $k>1$. 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).

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 (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 rectangles are not similar because side ratios are not equal: 2/3 ≠ 4/5 (or checking pairs: width 2 to 3 is 3/2, length 4 to 5 is 5/4, unequal), so choice C is correct. A common error is claiming all rectangles are similar, but only if proportions match (like squares), or thinking translations make them similar without dilation for size change. To check similarity: (1) measure corresponding sides (assume width 2 to 3, length 4 to 5), (2) calculate ratios (3/2=1.5, 5/4=1.25), (3) verify equal (no, not similar), (4) angles are all 90° but sides not proportional. Mistakes include claiming proportional when ratios differ or thinking dilations change angles, which they don't.

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. 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 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 (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.