This question tests understanding that rotations, reflections, and translations preserve line segment lengths—rigid transformations move segments without stretching or compressing. Rigid transformations (rotation, reflection, translation) preserve distances: segment AB with length $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ transforms to A'B' with equal length $d' = d$. Example: AB from (1,2) to (4,6) has length $\sqrt{9+16} = 5$, translation by (+3,+2) gives A'(4,4) to B'(7,8) with length $\sqrt{9+16} = 5$ unchanged. Property holds for all three transformations (move/flip/turn but don't resize). For this specific translation, original AB from (1,2) to (4,6) has length $\sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9+16} = 5$, and after adding (+3,+2), A'(4,4) to B'(7,8) has length $\sqrt{(7-4)^2 + (8-4)^2} = \sqrt{9+16} = 5$, confirming preservation. The correct answer is 5, as the translation shifts the position but keeps the distance between points the same. Common errors include miscalculating the distance as $\sqrt{ (4+1)^2 + (6+2)^2 } = \sqrt{25+64} = \sqrt{89}$ or confusing with vector addition leading to wrong lengths like 7 or 10. To verify: (1) calculate original length using distance formula, (2) apply translation by adding (3,2) to each coordinate, (3) calculate image length with new points, (4) compare—they are equal. All rigid transformations preserve length because they are isometries, maintaining distances while changing position or orientation but not size.

This question tests understanding that rotations, reflections, and translations preserve line segment lengths—rigid transformations move segments without stretching or compressing. Rigid transformations (rotation, reflection, translation) preserve distances: segment AB with length d=√((x₂-x₁)²+(y₂-y₁)²) transforms to A'B' with equal length d'=d. Example: AB from (1,2) to (4,6) has length √(9+16)=5, translation by (+3,+2) gives A'(4,4) to B'(7,8) with length √(9+16)=5 unchanged. Property holds for all three transformations (move/flip/turn but don't resize). For this reflection over the x-axis, original ST length is √[(6-0)² + (8-0)²] = √[36+64] = 10, and after reflection, S'(0,0) to T'(6,-8) has length √[(6-0)² + (-8-0)²] = √[36+64] = 10, confirming preservation. The length is 10 units, unchanged. A common error is calculating only partial differences, like √[36+16]=√52 or something leading to 14 if adding wrong. To verify: (1) calculate original length, (2) negate y-coordinates, (3) calculate image length, (4) compare—equal. All rigid transformations preserve length because they maintain distances (isometries: distance-preserving transformations)—the segment flips but not size.

This question tests understanding that rotations, reflections, and translations preserve line segment lengths—rigid transformations move segments without stretching or compressing. Rigid transformations (rotation, reflection, translation) preserve distances: segment AB with length d=√((x₂-x₁)²+(y₂-y₁)²) transforms to A'B' with equal length d'=d. Example: AB from (1,2) to (4,6) has length √(9+16)=5, translation by (+3,+2) gives A'(4,4) to B'(7,8) with length √(9+16)=5 unchanged. Property holds for all three transformations (move/flip/turn but don't resize). This general question confirms that translations, rotations, and reflections all preserve lengths, as they are rigid motions. This verifies the concept, as the correct answer is all of the above. A common error is thinking only one preserves length (e.g., claiming reflections change it), confusing with non-rigid like dilations. Verifying: (1) recall definitions, (2) apply each transformation to a sample segment, (3) check lengths equal, (4) confirm for all. All rigid transformations preserve length because they maintain distances (isometries: distance-preserving transformations)—segment moves position/orientation but not size. Common errors: assuming non-Euclidean metrics or misapplying rules leading to false length changes.

This question tests understanding that rotations, reflections, and translations preserve line segment lengths—rigid transformations move segments without stretching or compressing. Rigid transformations (rotation, reflection, translation) preserve distances: segment AB with length d=√((x₂-x₁)²+(y₂-y₁)²) transforms to A'B' with equal length d'=d. Example: AB from (1,2) to (4,6) has length √(9+16)=5, translation by (+3,+2) gives A'(4,4) to B'(7,8) with length √(9+16)=5 unchanged. Property holds for all three transformations (move/flip/turn but don't resize). For this 180° rotation, original LM length is √[(6-1)² + (4-4)²] = √[25+0] = 5, and after rotation, L'(-1,-4) to M'(-6,-4) has length √[(-6 - (-1))² + (-4 - (-4))²] = √[25+0] = 5, verifying preservation. The length is 5 units, as rotations keep distances. A common error is doubling differences, leading to 10 units. To verify: (1) calculate original length, (2) apply (x,y) to (-x,-y), (3) calculate image length, (4) compare—equal. All rigid transformations preserve length because they maintain distances (isometries: distance-preserving transformations)—the segment turns but not size.

This question tests understanding that rotations, reflections, and translations preserve line segment lengths—rigid transformations move segments without stretching or compressing. Rigid transformations (rotation, reflection, translation) preserve distances: segment AB with length d=√((x₂-x₁)²+(y₂-y₁)²) transforms to A'B' with equal length d'=d. Example: AB from (1,2) to (4,6) has length √(9+16)=5, translation by (+3,+2) gives A'(4,4) to B'(7,8) with length √(9+16)=5 unchanged. Property holds for all three transformations (move/flip/turn but don't resize). For this specific 90° counterclockwise rotation about the origin, original EF from (2,1) to (2,6) has length √((2-2)²+(6-1)²)=√(0+25)=5, and after rotation, E' is (-1,2) and F' is (-6,2), with length √((-6-(-1))²+(2-2)²)=√(25+0)=5. This verifies the length preservation, as the correct answer is 5 units, matching the unchanged distance. A common error is claiming rotation changes the length (e.g., 5 becomes 4 or 6 from misrotation), using wrong distance like |6-1| + |2-2| = 5 but altering to 10, or incorrect rotation rule like clockwise giving wrong points then wrong length. Verifying: (1) calculate original length with distance formula, (2) apply rotation using (x,y) to (-y,x), (3) calculate image length with the same formula, (4) compare (should equal). All rigid transformations preserve length because they maintain distances (isometries: distance-preserving transformations)—segment moves position/orientation but not size.

This question tests understanding that rotations, reflections, and translations preserve line segment lengths—rigid transformations move segments without stretching or compressing. Rigid transformations (rotation, reflection, translation) preserve distances: segment AB with length d=√((x₂-x₁)²+(y₂-y₁)²) transforms to A'B' with equal length d'=d. For GH from (-3,1) to (1,4), original length √((1 - (-3))² + (4-1)²) = √(16+9) = 5; after translation (-2,+5), G'(-5,6) to H'(-1,9), length √((-1 - (-5))² + (9-6)²) = √(16+9) = 5. The true statement is that length stays the same because translations preserve distance. Errors like claiming length becomes negative from the -2 or changes only for diagonal segments ignore preservation property. To verify: (1) calculate original length, (2) add translation vector to coordinates, (3) calculate image length, (4) compare—identical. All rigid transformations are isometries, preserving distances while shifting position.

This question tests understanding that rotations, reflections, and translations preserve line segment lengths—rigid transformations move segments without stretching or compressing. Rigid transformations (rotation, reflection, translation) preserve distances: segment AB with length d=√((x₂-x₁)²+(y₂-y₁)²) transforms to A'B' with equal length d'=d. Example: AB from (1,2) to (4,6) has length √(9+16)=5, translation by (+3,+2) gives A'(4,4) to B'(7,8) with length √(9+16)=5 unchanged. Property holds for all three transformations (move/flip/turn but don't resize). For this 90° counterclockwise rotation, original EF length is √[(2-2)² + (4 - (-1))²] = √[0+25] = 5, and after rotation, E'(1,2) to F'(-4,2) has length √[(-4-1)² + (2-2)²] = √[25+0] = 5, verifying preservation. The length is 5 units, unchanged by rotation. A common error is misapplying rotation rules, leading to wrong points and claiming 3 or 6 units. To verify: (1) calculate original length, (2) apply rotation using (x,y) to (-y,x), (3) calculate image length, (4) compare—equal. All rigid transformations preserve length because they maintain distances (isometries: distance-preserving transformations)—the segment turns but not size.

This question tests understanding that rotations, reflections, and translations preserve line segment lengths—rigid transformations move segments without stretching or compressing. Rigid transformations (rotation, reflection, translation) preserve distances: segment AB with length d=√((x₂-x₁)²+(y₂-y₁)²) transforms to A'B' with equal length d'=d. For QR from (-4,-2) to (-1,2), original length √((-1 - (-4))² + (2 - (-2))²) = √(9+16) = 5; after translation (+5,-3) to (1,-5) and (4,-1), then y-axis reflection to (-1,-5) and (-4,-1), length √((-4 - (-1))² + (-1 - (-5))²) = √(9+16) = 5. Correct length is 5, as multiple transformations preserve distance. Error: incorrect order or rules, leading to wrong points and lengths like 8. Verify: (1) original length, (2) apply translation then reflection, (3) image length, (4) same. Isometries compose to preserve lengths.

This question tests understanding that rotations, reflections, and translations preserve line segment lengths—rigid transformations move segments without stretching or compressing. Rigid transformations (rotation, reflection, translation) preserve distances: segment AB with length d=√((x₂-x₁)²+(y₂-y₁)²) transforms to A'B' with equal length d'=d. For NP from (2,7) to (6,2), original length √((6-2)² + (2-7)²) = √(16+25) = √41; after y=x reflection, N'(7,2) to P'(2,6), length √((2-7)² + (6-2)²) = √(25+16) = √41. Correct length is √41, verified by swapped coordinates yielding same differences. Error: confusing with distance as sum, getting 9 instead. Verify by: (1) original distance, (2) swap x and y, (3) new distance, (4) equal. Isometries flip but keep size.

This question tests understanding that rotations, reflections, and translations preserve line segment lengths—rigid transformations move segments without stretching or compressing. Rigid transformations (rotation, reflection, translation) preserve distances: segment AB with length d=√((x₂-x₁)²+(y₂-y₁)²) transforms to A'B' with equal length d'=d. Example: AB from (1,2) to (4,6) has length √(9+16)=5, translation by (+3,+2) gives A'(4,4) to B'(7,8) with length √(9+16)=5 unchanged. Property holds for all three transformations (move/flip/turn but don't resize). The student's claim is incorrect because rotations preserve lengths regardless of direction; for example, rotating (0,0) to (3,4) by 90° gives (0,0) to (-4,3), both lengths √[9+16]=5. The best correction is that rotations are rigid transformations that preserve length. A common error is thinking direction changes length, like claiming it doubles. To verify the concept: (1) calculate original length, (2) apply rotation rules, (3) calculate image length, (4) compare—always equal for rigid motions. All rigid transformations preserve length because they maintain distances (isometries: distance-preserving transformations)—the segment turns but not size.