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.

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. 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. For segment CD from (-2,5) to (4,5), original length is √((4 - (-2))² + (5-5)²) = √36 = 6; after reflection over y-axis, C'(2,5) to D'(-4,5), length √((-4-2)² + (5-5)²) = √36 = 6. The correct statement is that the length stays the same, as reflection flips the segment but preserves distances. A common error is thinking reflection changes length due to negative x-values, but coordinates' signs don't affect distance calculation. To verify: (1) calculate original length with distance formula, (2) apply reflection by negating x-coordinates, (3) calculate image length, (4) compare—they match. All rigid transformations preserve length because they are isometries: distance-preserving transformations that move the segment's position or orientation but not its size. Another error is assuming horizontal segments halve or double when reflected, ignoring that only position 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. For LM from (-1,-4) to (3,-4), original length √((3 - (-1))² + (-4 - (-4))²) = √16 = 4; after 180° rotation, L'(1,4) to M'(-3,4), length √((-3-1)² + (4-4)²) = √16 = 4. The correct length is 4, as rotation inverts coordinates but preserves distance. Errors include doubling to 8 by adding differences or claiming negative length from coordinates. To verify: (1) calculate original, (2) apply (x,y) → (-x,-y), (3) calculate new length, (4) compare—same. Isometries ensure length preservation across turns.

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 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 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. 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 GH length is √[(3 - (-1))² + (0 - (-3))²] = √[16+9] = 5, and after reflection, G'(-1,3) to H'(3,0) has length √[(3 - (-1))² + (0-3)²] = √[16+9] = 5, confirming the same. The length is 5 units, as expected for rigid transformations. A common error is using taxicab distance like |3 - (-1)| + |0 - (-3)| = 7, picking 7 units. To verify: (1) calculate original length, (2) apply reflection by negating y-coordinates, (3) calculate image length, (4) compare—identical. All rigid transformations preserve length because they maintain distances (isometries: distance-preserving transformations)—the segment flips but not size.