Understand Line Segment Transformation Properties - 8th Grade Math

Segment $\overline{LM}$ has endpoints $L(1,4)$ and $M(6,4)$ . It is rotated $180^\circ$ about the origin to form $\overline{L'M'}$ . What is the length of $\overline{L'M'}$ ?
Answer: 5 units 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.
A line segment AB is reflected across line j, then the resulting segment A'B' is translated 5 units left to form segment A''B''. If the original segment AB has length $2\sqrt{5}$ units, what is the length of segment A''B''?
Answer: $2\sqrt{5}$ units because both reflection and translation preserve segment lengths throughout the sequence When you encounter transformation problems, the key concept to remember is that certain transformations preserve distance while others don't. Reflections and translations are both rigid transformations, meaning they move figures without changing their size or shape. Let's trace through this problem step by step. The original segment AB has length $2\sqrt{5}$ units. When AB is reflected across line j, the resulting segment A'B' maintains exactly the same length as the original because reflection acts like flipping the segment over a mirror line—the distance between the endpoints stays identical. Then, when A'B' is translated 5 units left to create A''B'', translation simply slides the entire segment to a new position without stretching or shrinking it. Looking at the wrong answers: Choice A incorrectly assumes that moving 5 units left somehow subtracts from the segment's length, confusing the direction of movement with the measurement of the segment itself. Choice C makes the opposite error, thinking the translation adds to the length. Choice D applies the Pythagorean theorem as if the transformations created some kind of right triangle, which completely misunderstands what's happening—we're measuring the length of the final segment, not the distance between original and final positions. The answer is B: $2\sqrt{5}$ units because both transformations preserve length. Study tip: Remember that reflections, rotations, and translations are rigid transformations that preserve all distances and angles. Only dilations (scaling) change the size of geometric figures.
Triangle PQR undergoes a 270° clockwise rotation about vertex P to form triangle PQ'R'. If side QR has length 12 units, which statement about the transformed triangle is correct?
Answer: Side Q'R' has length 12 units because rotations about any point preserve segment lengths When you encounter questions about geometric transformations, focus on what properties are preserved versus what changes during the transformation. Rotations are rigid transformations, meaning they preserve all distances and angle measures regardless of the rotation angle or center point. When triangle PQR rotates 270° clockwise about vertex P, every point moves along a circular arc, but the distances between any two points remain exactly the same. Since side QR has length 12 units in the original triangle, the corresponding side Q'R' in the rotated triangle must also have length 12 units. Let's examine why the other options are incorrect. Choice A suggests the length is preserved "only because" the rotation is about vertex P, but this misses the key point—rotations preserve distances regardless of which point serves as the center. Choice B incorrectly claims that Q'R' corresponds to side PQ, when actually Q'R' corresponds to the original side QR. The lengths would only match if PQ happened to equal QR. Choice C reflects a common misconception that rotations stretch figures like some other transformations do, but rotations never change distances—they only change position and orientation. The correct answer is D because rotations preserve segment lengths no matter what angle of rotation or center point you use. This is a fundamental property of rigid transformations. Study tip: Remember that rigid transformations (rotations, reflections, translations) always preserve distances and angles. Only non-rigid transformations like dilations change lengths.
Rectangle ABCD is translated to form rectangle A'B'C'D'. If diagonal AC has length 10 cm, what must be true about diagonal A'C' in the translated rectangle?
Answer: Diagonal A'C' has length 10 cm because translations preserve all segment lengths within figures When you encounter questions about transformations like translations, the key concept to remember is that translations are rigid transformations—they move figures without changing their size or shape. A translation slides every point of a figure the same distance in the same direction. Since rectangle ABCD is simply moved to a new position to form A'B'C'D', all the measurements within the rectangle remain exactly the same. This includes side lengths, angles, and crucially, diagonal lengths. If diagonal AC measures 10 cm in the original rectangle, then diagonal A'C' must also measure 10 cm because translations preserve all distances and measurements within figures. Let's examine why the other answers are incorrect. Choice A suggests the diagonal length depends on the translation vector's direction and distance, but this confuses the movement of the rectangle's position with changes to its internal measurements—translations don't alter any lengths within the figure. Choice B reaches the correct conclusion but gives an incomplete reason; while the rectangle's shape does determine diagonal length, this doesn't explain why translation preserves that measurement. Choice C contains a fundamental misconception, claiming translations change internal structure, which contradicts the definition of rigid transformations. The correct answer is D because it accurately states both the result (10 cm diagonal) and the complete reasoning (translations preserve all segment lengths). Study tip: Remember that translations, rotations, and reflections are all rigid transformations—they preserve size, shape, and all internal measurements. Only dilations change lengths.
Segment $\overline{LM}$ has endpoints $L(5,-1)$ and $M(2,-5)$ . It is reflected over the $x$ -axis to form $\overline{L'M'}$ . What is the length of $\overline{L'M'}$ ?
Answer: $\sqrt{25}$ units 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 reflection over the x-axis, original LM from (5,-1) to (2,-5) has length √((2-5)²+(-5-(-1))²)=√(9+16)=5, and after reflection, L' is (5,1) and M' is (2,5), with length √((2-5)²+(5-1)²)=√(9+16)=5. This verifies the length preservation, as the correct answer is √25 units (which is 5), matching the unchanged distance. A common error is claiming reflection changes length (e.g., to √41 from wrong dy), using |2-5| + | -5 - (-1)|=7, or wrong reflection rule like over y-axis giving incorrect points then wrong length. Verifying: (1) calculate original length with distance formula, (2) apply reflection by negating y-coordinates, (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.
Segment $\overline{QR}$ has endpoints $Q(-4,0)$ and $R(-1,4)$ . It is translated by $(+5,-3)$ and then reflected over the $y$ -axis to form $\overline{Q''R''}$ . What is the length of $\overline{Q''R''}$ ?
Answer: $5$ units 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 these combined transformations, original QR from (-4,0) to (-1,4) has length √((-1-(-4))²+(4-0)²)=√(9+16)=5, after translation to (1,-3) and (4,1), then reflection to (-1,-3) and (-4,1), with length √((-4-(-1))²+(1-(-3))²)=√(9+16)=5. This verifies the length preservation, as the correct answer is 5 units, matching the unchanged distance. A common error is misapplying steps (e.g., reflecting first, getting wrong points leading to √5 or 7), or using addition for distance giving 7. Verifying: (1) calculate original length with distance formula, (2) apply translation then reflection step-by-step, (3) calculate final image length, (4) compare (should equal). All rigid transformations preserve length because they maintain distances (isometries: distance-preserving transformations)—segment moves position/orientation but not size.
Line segment MN is reflected across the y-axis to create line segment M'N'. If the length of MN is 8 units, which statement about this transformation is correct?
Answer: The length of M'N' is 8 units because reflections preserve distances between points Under reflections, line segments map to line segments of the same length. This is a fundamental property of rigid transformations. Choice B gives the correct length but incorrect reasoning about perpendicular bisectors. Choice C incorrectly doubles the length by confusing the reflected segment with the distance across the reflection line. Choice D incorrectly suggests that segment length depends on position relative to the reflection line.
Segment $LM$ has endpoints $L(-1,-4)$ and $M(3,-4)$ . It is rotated $180^\circ$ about the origin to form $L'M'$ . What is the length of $L'M'$ ?
Answer: $4$ 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.
Segment $\overline{CD}$ has endpoints $C(-2,1)$ and $D(4,1)$ . It is reflected over the $y$ -axis to form $\overline{C'D'}$ . How does the length of $\overline{C'D'}$ compare to the length of $\overline{CD}$ ?
Answer: It is the same length. 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 y-axis, original CD length is √[(4 - (-2))² + (1-1)²] = √[36+0] = 6, and after reflection, C'(2,1) to D'(-4,1) has length √[(-4-2)² + (1-1)²] = √[36+0] = 6, confirming equal lengths. The lengths are the same, as reflections preserve distances. A common error is thinking negative coordinates shorten the segment, like claiming it's shorter. 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 maintain distances (isometries: distance-preserving transformations)—the segment flips but not size.
A student says: "If you rotate a segment $90^\circ$ , its length might change because it points in a different direction." Which statement best corrects the student?
Answer: The student is incorrect; rotations are rigid transformations that preserve length. 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.
A line segment on grid paper has endpoints $G(-1,-3)$ and $H(3,0)$ . It is reflected over the $x$ -axis to form $\overline{G'H'}$ . What is the length of $\overline{G'H'}$ ?
Answer: $5$ units 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.
In the coordinate plane shown, line segment PQ is translated 4 units right and 3 units down to form line segment P'Q'. What is the relationship between the lengths of PQ and P'Q'?
Answer: P'Q' has the same length as PQ because translations preserve segment lengths Under translations, line segments map to line segments of the same length. This is true regardless of the direction or magnitude of the translation vector. Choice A incorrectly relates segment length to distance from origin. Choice B incorrectly suggests translations compress figures. Choice D incorrectly suggests the direction of translation affects segment length preservation.
Segment $\overline{EF}$ has endpoints $E(2,1)$ and $F(2,6)$ . It is rotated $90^\circ$ counterclockwise about the origin to form $\overline{E'F'}$ . What is the length of $\overline{E'F'}$ ?
Answer: 5 units 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.
Segment $\overline{NP}$ has endpoints $N(1,4)$ and $P(6,2)$ . It is rotated $90^\circ$ counterclockwise about the origin to form $\overline{N'P'}$ . What is the length of $\overline{N'P'}$ ?
Answer: $\sqrt{29}$ units 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 NP from (1,4) to (6,2) has length √((6-1)²+(2-4)²)=√(25+4)=√29, and after rotation, N' is (-4,1) and P' is (-2,6), with length √((-2-(-4))²+(6-1)²)=√(4+25)=√29. This verifies the length preservation, as the correct answer is √29 units, matching the unchanged distance. A common error is claiming rotation changes length (e.g., to √34 from miscalculation), using wrong rotation like (-x,y) giving wrong points, or taxicab distance |5|+| -2|=7 not sqrt. 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.
On a coordinate plane, segment $\overline{AB}$ has endpoints $A(1,2)$ and $B(4,6)$ . The segment is translated by $(+3,+2)$ to form $\overline{A'B'}$ . What is the length of $\overline{A'B'}$ ?
Answer: 5 units 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 translation, original AB length is √[(4-1)² + (6-2)²] = √[9+16] = 5, and after translation, A'(4,4) to B'(7,8) has length √[(7-4)² + (8-4)²] = √[9+16] = 5, confirming preservation. The correct answer is 5 units, as the length remains unchanged. A common error is miscalculating the distance by adding instead of using the formula, like claiming 7 units from incorrect addition of differences. To verify: (1) calculate original length with distance formula, (2) apply translation by adding (+3,+2) to each coordinate, (3) calculate image length with the same formula, (4) compare—they are equal. All rigid transformations preserve length because they maintain distances (isometries: distance-preserving transformations)—the segment moves position but not size.
Segment $NP$ has endpoints $N(2,7)$ and $P(6,2)$ . It is reflected over the line $y=x$ to form $N'P'$ . What is the length of $N'P'$ ?
Answer: $\sqrt{41}$ 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.
Segment $\overline{ST}$ has endpoints $S(0,0)$ and $T(6,8)$ . It is reflected over the $x$ -axis to form $\overline{S'T'}$ . What is the length of $\overline{S'T'}$ ?
Answer: $10$ units 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.
Triangle ABC is rotated 90° clockwise about point P to form triangle A'B'C'. If side AB has length 7 cm and side BC has length 5 cm, what can be concluded about the corresponding sides in triangle A'B'C'?
Answer: Side A'B' has length 7 cm and side B'C' has length 5 cm Under rotations, line segments map to line segments of the same length. Therefore, AB maps to A'B' with the same length (7 cm), and BC maps to B'C' with the same length (5 cm). Choice B incorrectly swaps the lengths. Choice C incorrectly suggests that the position of the rotation point affects lengths. Choice D incorrectly applies properties of similarity transformations rather than rigid transformations.
Segment $\overline{EF}$ has endpoints $E(2,-1)$ and $F(2,4)$ . It is rotated $90^\circ$ counterclockwise about the origin to form $\overline{E'F'}$ . What is the length of $\overline{E'F'}$ ?
Answer: 5 units 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.
A student draws segment $QR$ from $Q(-4,-2)$ to $R(-1,2)$ . Then the student translates it by $(+5,-3)$ and reflects the result over the $y$ -axis to get $Q''R''$ . What is the length of $Q''R''$ ?
Answer: $5$ 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.

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