8.G.A.1: Properties of Rotations, Reflections, and Translations
In plain English: 8.G.A.1 is the Common Core grade 8 math standard that asks students to verify by experiment what rotations, reflections and translations do to figures, using tracing paper, grids or geometry software. Students check that these moves take lines to lines, keep lengths and angle measures the same, and keep parallel lines parallel. It opens the Grade 8 Math unit on congruence (figures with the same size and shape).
Verify experimentally the properties of rotations, reflections, and translations:
a.Lines are taken to lines, and line segments to line segments of the same length.
b.Angles are taken to angles of the same measure.
c.Parallel lines are taken to parallel lines.
Common Core State Standards for Mathematics · Domain: Geometry (G) · Cluster: Understand congruence and similarity using physical models, transparencies, or geometry software. Also written as 8.G.1 · Official standard
A transformation is a rule that moves every point of a figure to a new position. This lesson studies three of them: a translation (a slide), a reflection (a flip over a line) and a rotation (a turn about a point). Together they are called rigid motions, because they move a figure without stretching, shrinking or bending it. The figure after the move is the image, and its points are named with a prime mark: A moves to A′, read "A prime".
The standard asks students to verify experimentally, which means to check by doing and measuring rather than by proof. Students move figures with tracing paper, transparencies or geometry software, then measure the original and the image with a ruler and a protractor, or count grid squares. They confirm three facts for every rigid motion: (a) lines go to lines, and line segments (parts of a line between two endpoints) go to segments of the same length; (b) angles go to angles of the same measure; (c) parallel lines go to parallel lines. These facts are the base for congruence in 8.G.A.2.
Learning Objectives
By the end of this lesson, students will be able to:
Carry out translations, reflections and rotations of figures with tracing paper and on a coordinate grid, and label images with prime marks
Verify by measuring that each rigid motion takes lines to lines and line segments to segments of the same length
Verify by measuring that each rigid motion takes angles to angles of the same measure
Verify that each rigid motion takes a pair of parallel lines to a pair of parallel lines
Tell apart what a rigid motion can change (position, the direction a side points, the order of the vertices, or corners) from what it keeps
Prior Knowledge Required
Students should already be comfortable with:
Drawing and naming points, lines, line segments, angles, and parallel lines (lines in a plane that never meet) 4.G.A.1
Measuring angles in whole-number degrees with a protractor 4.MD.C.6
Lines of symmetry: a fold line that makes the two halves of a figure match 4.G.A.3
Plotting points in all four quadrants of the coordinate plane 6.NS.C.8
Finding the length of a horizontal or vertical segment from the coordinates of its endpoints 6.G.A.3
Give each student a sheet of tracing paper and a ruler.
Warm-Up Prompt
"Draw a segment 6 cm long on your tracing paper and label its ends P and Q. Slide the paper across your desk, then flip it over, then turn it. After each move, measure PQ again. Is it still straight? Is it still 6 cm long? What did change?"
Students find that PQ stays straight and stays 6 cm long after every move; only its place on the desk changed. Write the three moves on the board as slide, flip and turn, and tell students that today they give them their math names and test angles and parallel lines the same way.
Direct Instruction20 minutes
Define each word as you demonstrate it with a transparency on the projector. Students copy each definition next to a quick sketch.
Translation: a slide. Every point moves the same distance in the same direction, for example 7 units left and 7 units down.
Reflection: a flip over a line, called the line of reflection. Each image point is the same distance from that line as its original point, on the other side, as if the line were a fold.
Rotation: a turn about a fixed point, called the center of rotation, through an angle such as 90° (a quarter turn) or 180° (a half turn), clockwise or counterclockwise. Every point stays the same distance from the center. To rotate with tracing paper, hold the center down with a pencil point and turn the paper.
How to verify: do the move, then measure the original and the image with a ruler and a protractor, or count grid squares for horizontal and vertical sides. Paper measurements can be off by a millimeter or a degree, so treat values that close as equal.
What the experiments show: for all three motions, lines go to lines, segments keep their length, angles keep their measure, and parallel lines stay parallel. The position of a figure can change, and so can the direction its sides point.
Translation (Diagram 1)
Triangle ABC has vertices A(1, 1), B(5, 1) and C(1, 4) on centimeter grid paper. Trace it, then slide the tracing paper 7 units left and 7 units down.
Equation: The image has A′(-6, -6), B′(-2, -6) and C′(-6, -3). Counting squares: AB = A′B′ = 4 and AC = A′C′ = 3. A ruler gives BC = B′C′ = 5 cm. A protractor gives 90° at A and A′, about 37° at B and B′, and about 53° at C and C′.
Reflection across the y-axis (Diagram 1)
Fold the grid paper along the y-axis and trace triangle ABC onto the other side.
Equation: Each point lands the same distance from the y-axis on the other side: A′(-1, 1), B′(-5, 1), C′(-1, 4). The sides still measure 4, 3 and 5, and the angles still measure 90°, about 37° and about 53°. One thing changes: going A to B to C turns counterclockwise, but going A′ to B′ to C′ turns clockwise, because the figure was flipped over.
Rotation 90° clockwise about the origin (Diagram 1)
Hold the tracing of triangle ABC down at the origin (0, 0) and turn the paper a quarter turn clockwise.
Equation: The image has A′(1, -1), B′(1, -5) and C′(4, -1). AB was horizontal and A′B′ is vertical, yet both are 4 units long. AC was vertical and A′C′ is horizontal, both 3 units long. The right angle is still a right angle, and the other two angles still measure about 37° and about 53°.
Parallel lines under a rotation (Diagram 2)
Line j passes through P(-6, 1) and Q(-2, 3). Line k passes through R(-6, -2) and S(-2, 0). Each line goes 2 across for every 1 up, so they never meet: they are parallel. Rotate both lines 90° counterclockwise about the origin.
Equation: The images pass through P′(-1, -6), Q′(-3, -2), R′(2, -6) and S′(0, -2). Both image lines go 1 left for every 2 up, so j′ and k′ are parallel too. Segment PQ goes 4 across and 2 up; P′Q′ goes 2 left and 4 up, the same grid steps turned a quarter turn, so the two segments are the same length.
On Diagram 1, point out that the three images sit in different quadrants and face different ways, yet every side and angle matches the original. On Diagram 2, have students lay a ruler across j′ and k′ at two places and check that the gap between them is the same.
Guided Practice15 minutes
Pairs work each problem on grid paper with tracing paper. One partner moves the figure and the other measures; they switch roles for the next problem.
Guided practice problems with answers
Problem
Answer
Segment FG runs from F(-3, 2) to G(3, 2). Translate it 2 units right and 4 units down. Where is the image, and how long is it?
F′(-1, -2) and G′(5, -2). It is still a horizontal segment, 6 units long
Angle HJK has H(0, 4), J(0, 0) and K(4, 4). Reflect it across the x-axis. What does the image angle measure?
H′(0, -4), J′(0, 0), K′(4, -4). JK cuts each grid square corner to corner, so both angles measure 45°, half of a right angle
A rectangle has vertices (1, 1), (4, 1), (4, 3) and (1, 3). Rotate it 180° about the origin. Compare the sides and angles.
Image vertices (-1, -1), (-4, -1), (-4, -3) and (-1, -3). The sides still measure 3 and 2 units and all four angles are still 90°
The horizontal lines through (0, 1) and (0, 4) are parallel. Rotate both 90° counterclockwise about the origin. What are the images?
The vertical lines through (-1, 0) and (-4, 0). They are still parallel, 3 units apart
Listen for students who count the translation from the wrong point, or who turn the tracing paper without holding the center still. Ask them to name the center or the direction before they move anything.
Independent Practice10-15 minutes
Students work alone, then compare answers with a partner.
Independent practice problems with answers
Problem
Answer
Segment LM runs from L(-4, 1) to M(-4, 6). Translate it 5 units right and 2 units up. Give the image endpoints and both lengths.
L′(1, 3) and M′(1, 8). LM = L′M′ = 5 units
A triangle has vertices (2, 0), (6, 0) and (2, 2). Reflect it across the x-axis. Which image angle is a right angle?
Image (2, 0), (6, 0), (2, -2). The angle at (2, 0) is still 90°. The points (2, 0) and (6, 0) do not move, because they are on the line of reflection
A trapezoid (a quadrilateral with one pair of parallel sides) has parallel sides 3 cm and 7 cm long. It is rotated 60° about one corner. What do you know about the images of those two sides?
They are still parallel, and they still measure 3 cm and 7 cm
An angle measures 118°. It is rotated 90° clockwise. What does the image angle measure?
118°. A rotation turns the angle, but the opening between its sides stays the same
In one sentence, explain why a translation cannot bend a segment into a curve.
Every point moves the same distance in the same direction, so points that were in a straight row are still in a straight row
Closure5 minutes
Exit ticket: (1) A 5 cm segment is reflected across a line. How long is its image? (5 cm.) (2) An angle of 72° is translated 3 units up. What does the image angle measure? (72°.) (3) In one sentence, describe how you would use tracing paper to check that a rotation keeps two parallel lines parallel.
Differentiation Strategies
For Struggling Students
Give figures whose vertices sit where grid lines cross, so students can count every horizontal and vertical length
Have students draw an arrow on the tracing paper before a translation, and a dot at the center before a rotation, so the move is visible
Use a recording frame for each move: "Side before ____, side after ____; angle before ____, angle after ____; parallel before? ____ after? ____"
For Advanced Students
Ask students to find a rotation that moves segment PQ in Diagram 2 onto a segment going 2 right and 4 down, and to name its center and angle
Have students investigate whether a rotation ever reverses the order of the vertices, and whether two reflections in a row do
Extension (beyond this standard): apply a dilation (a scaling from a center point, studied in 8.G.A.3 and 8.G.A.4) with scale factor 2 and record which of the three properties still hold
Assessment Guidance
What to Look For
A complete answer names the motion with its details (the direction and distance of a translation, the line of a reflection, the center, angle and direction of a rotation), lists the image points with prime marks, and backs each claim with a measurement: a side length, an angle measure, or matching grid steps for parallel lines. Watch for students who say a rotation made a side longer because it now points a different way, who reflect across the wrong axis, and who treat values 1 mm or 1° apart as different.
02
Classroom Activities
3 Activities
1
Tracing-Paper Lab
20 minPairs
Pairs draw trapezoid WXYZ with W(1, 1), X(7, 1), Y(5, 4) and Z(2, 4) on centimeter grid paper, move it three ways with tracing paper, and record measurements of the original and each image in a table.
The Three Moves
Translation 8 units left
Reflection across the x-axis
Rotation 180° about the origin
Procedure
Trace WXYZ and its labels, carry out one move, and copy the image onto the grid with prime labels
Measure all four sides with a ruler to the nearest millimeter and all four angles with a protractor
Check that the images of the parallel sides WX and ZY are still parallel by counting grid steps along each
Repeat for the other two moves and compare the four rows of the table
Every figure: WX = 6 cm, XY ≈ 3.6 cm, YZ = 3 cm, ZW ≈ 3.2 cm; angles W ≈ 72°, X ≈ 56°, Y ≈ 124°, Z ≈ 108°; the images of WX and ZY are horizontal, so they stay parallel
Discussion Questions
Only one image lists its vertices W′, X′, Y′, Z′ in clockwise order. Which move made it, and why?
Only the translation image still has its long side at the bottom. What did the other two moves do to that side?
Your four angles add to about 360° each time. Why is that a useful check on your protractor readings?
Modification for Distance Learning
Students draw WXYZ in free geometry software, use its translate, reflect and rotate tools, and read the lengths and angles with the software's measuring tools.
2
Always, Sometimes or Never?
15 minPairs
Pairs get 8 statement cards. For each card they test examples with tracing paper and grid paper, sort it into Always, Sometimes or Never, and write one example that supports their choice.
The 8 Cards
Card A: A rotation takes a segment to a segment of the same length.
Card B: A rotation takes a horizontal line to a horizontal line.
Card C: A translation takes two parallel lines to two lines that meet.
Card D: A reflection changes the measure of an angle.
Card E: A translation takes a segment to a segment that points in the same direction.
Card F: A rotation moves every point of a figure.
Card G: A reflection turns a counterclockwise order of vertices into a clockwise order.
Card H: A figure and its image under a translation have the same size and shape.
Answer Key
Always: A, E, G, H
Sometimes: B (a half turn keeps a horizontal line horizontal; a quarter turn makes it vertical) and F (a point that is the center of rotation stays where it is)
Never: C, D
Discussion Questions
For each Sometimes card, give one example where it is true and one where it is false.
Which cards did you decide by testing, and which by thinking about what the motion does?
3
Parallel Lines Challenge
20 minGroups of 3
Each group draws line u through A(-6, -1) and B(3, 2), line v through C(-6, -4) and D(3, -1), and the vertical line t through (-3, 5) and (-3, -5). Line t is a transversal: a line that crosses two or more other lines. It meets u at E(-3, 0) and v at F(-3, -3). Each group then gets one motion card, moves the whole drawing, and checks the parallel lines and the angles.
Setup Checks
u and v each go 9 across and 3 up between their two marked points, so they are parallel
Mark the angle above u and to the right of t at E, and the matching angle at F. Each measures about 72°
The 4 Motion Cards
Card 1: Translate 2 units right and 3 units up
Card 2: Reflect across the x-axis
Card 3: Rotate 90° clockwise about the origin
Card 4: Rotate 180° about the origin
Answer Key
Card 1: A′(-4, 2), B′(5, 5), C′(-4, -1), D′(5, 2), E′(-1, 3), F′(-1, 0); u′ and v′ each go 9 across and 3 up
Card 2: A′(-6, 1), B′(3, -2), C′(-6, 4), D′(3, 1), E′(-3, 0), F′(-3, 3); u′ and v′ each go 9 across and 3 down
Card 3: A′(-1, 6), B′(2, -3), C′(-4, 6), D′(-1, -3), E′(0, 3), F′(-3, 3); u′ and v′ each go 3 across and 9 down
Card 4: A′(6, 1), B′(-3, -2), C′(6, 4), D′(-3, 1), E′(3, 0), F′(3, 3); u′ and v′ each go 9 across and 3 up, from B′ to A′ and from D′ to C′
On every card, u′ and v′ are parallel, and the marked angles at E′ and F′ still measure about 72°
Discussion Questions
Only Card 3 turns t into a horizontal line. What do the other three cards do to t?
On Card 2, E did not move. Why not?
No group found image lines that meet. What would it take to make u′ and v′ meet?
Challenge Variation
Groups find a single translation that moves line u exactly onto line v, then test it with tracing paper.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: One Triangle and Its Three Images
Triangle ABC with A(1, 1), B(5, 1) and C(1, 4), drawn to scale, and its images after a translation 7 units left and 7 units down, a reflection across the y-axis, and a rotation 90° clockwise about the origin. Every image has sides 4, 3 and 5 units and angles of 90°, about 37° and about 53°, like the original.
Diagram 2: A Rotation Keeps Parallel Lines Parallel
Parallel lines j (through P and Q) and k (through R and S), and their images j′ and k′ after a rotation of 90° counterclockwise about the origin, drawn to scale. The original lines each go 2 across for every 1 up; the images each go 1 left for every 2 up, so they are parallel too.
04
Homework Assignment
~30 min
8.G.A.1 Homework: Testing Rigid Motions
Directions: Use grid paper and tracing paper for every problem. Label every image point with a prime mark, and back up each answer with a measurement or a count of grid squares.
Part 1: Move and Measure (Problems 1-3)
Triangle RST has vertices R(-6, 2), S(-2, 2) and T(-2, 5). (a) Translate it 6 units right and 3 units down and list the image vertices. (b) Find RS, ST, R′S′ and S′T′ by counting squares. (c) Measure angle S and angle S′ with a protractor.
Triangle JKL has vertices J(2, 1), K(6, 1) and L(2, 5). (a) Reflect it across the y-axis and list the image vertices. (b) Measure the three angles of each triangle. (c) Going J to K to L turns counterclockwise. Which way does J′ to K′ to L′ turn?
Segment UV runs from U(2, 1) to V(5, 3). (a) Rotate it 90° counterclockwise about the origin and give the endpoints of U′V′. (b) Describe the grid steps from U to V and from U′ to V′. (c) Explain why the two segments have the same length.
Part 2: Angles and Parallel Lines (Problems 4-6)
Line f passes through (0, 2) and (3, 3), and line g passes through (0, -1) and (3, 0). (a) Explain why f and g are parallel. (b) Reflect both lines across the x-axis and give the images of the four points. (c) Show that f′ and g′ are parallel.
Nadia traces angle PQR, which measures 64°, and rotates it 90° clockwise. She says the image measures 26° "because a quarter turn takes 90° away from the angle". (a) What does the image measure? (b) Where did 26 come from? (c) Describe a quick experiment that would show her the correct answer.
A floor tile has the shape of parallelogram EFGH with E(0, 0), F(4, 0), G(6, 3) and H(2, 3). (a) Rotate it 180° about the origin and list the image vertices. (b) Name the two pairs of parallel sides in the image. (c) Angle E measures about 56°. What does angle E′ measure, and why did E not move?
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Images
Every image drawn in the right place with prime labels
One image misplaced or labels missing
Images missing or wrong
Measurements
Lengths and angles measured or counted for the original and the image
Some measurements missing
No measurements
Parallel Lines
Parallel images shown with matching grid steps
Says "parallel" without evidence
Missing or incorrect
Explanations
Clear reasons in words for each claim, including the error in Problem 5
Reasons given for some claims
No reasons given
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order, using grid paper and tracing paper when they help. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Which move is a translation?
Answer: A
In choice A every point slides the same distance in the same direction, which is what a translation does. Choice B is a rotation and choice C is a reflection, the other two rigid motions. Choice D stretches the figure to twice its size, so it is not a rigid motion at all.
Question 2 of 20 · Multiple Choice
A segment 7 cm long is rotated 90° counterclockwise about one of its endpoints. How long is the image?
Answer: C
A rotation keeps every segment the same length, so the image is 7 cm long; it only points a different way. Choice D is the straight-line distance the free endpoint moved (about 9.9 cm), not the length of the image. Choices A and B halve or double the length, but a rotation does not change size.
Question 3 of 20 · Multiple Choice
Angle DEF measures 48°. It is reflected across a line. What does angle D′E′F′ measure?
Answer: D
A reflection keeps angle measures, so the image measures 48°. Choice A is 90° - 48°, choice B is 180° - 48°, and choice C doubles the angle; none of these happens under a reflection.
Question 4 of 20 · Multiple Choice
Point K(5, -3) is reflected across the y-axis. Where is K′?
Answer: A
K is 5 units right of the y-axis, so K′ is 5 units left of it at the same height: (-5, -3). Choice B reflects across the x-axis instead. Choice C is where a half turn about the origin would send K. Choice D swaps the two coordinates.
Question 5 of 20 · Multiple Choice
Point M(-4, 6) is translated 7 units right and 2 units down. Where is M′?
Answer: D
Right 7 changes x from -4 to 3, and down 2 changes y from 6 to 4, so M′ is (3, 4). Choice A moves left instead of right. Choice B moves up instead of down. Choice C swaps the two distances: 2 right and 7 down.
Question 6 of 20 · Multiple Choice
Point N(2, 5) is rotated 180° about the origin. Where is N′?
Answer: B
A half turn about the origin sends N to the opposite side of the origin, the same distance away: (-2, -5). Choice C is a quarter turn clockwise. Choices A and D are reflections across the y-axis and the x-axis.
Question 7 of 20 · Multiple Choice
Lines p and q are parallel. Both are reflected across the same line. What is true about p′ and q′?
Answer: C
Rigid motions take parallel lines to parallel lines, so p′ and q′ are parallel to each other, whatever the line of reflection is. Choice D is the error of testing only one case, such as a reflection across the y-axis, and deciding the property depends on it. Choices A and B are not true in general.
Question 8 of 20 · Multiple Choice
Segment GH is vertical and 5 units long. It is rotated 180° about the origin. What is true about G′H′?
Answer: A
A half turn points every segment in the opposite direction, so a vertical segment stays vertical, and a rotation keeps its length of 5 units. Choice B is what a quarter turn would do. Choices C and D double the length, as if the 180° made the segment grow.
Question 9 of 20 · Multiple Choice
A triangle has angles of 35°, 65° and 80°. It is rotated 90° clockwise. What does the image of the 65° angle measure?
Answer: D
A rotation keeps every angle measure, so the image angle is 65°. Choice A adds the 90° turn to the angle. Choice B subtracts the angle from 90°. Choice C subtracts it from 180°.
Question 10 of 20 · Multiple Choice
The segment from (1, 2) to (1, 8) is translated 4 units right and 2 units down. Which describes the image?
Answer: D
Each endpoint moves 4 right and 2 down: (1, 2) goes to (5, 0) and (1, 8) goes to (5, 6). Both segments are 8 - 2 = 6 units long. Choice A moves up instead of down. Choice B swaps the two distances and moves up. Choice C reads the length as 8, the y-coordinate of the top endpoint.
Question 11 of 20 · Multiple Choice
Line r is the x-axis and line s is the horizontal line through (0, 2), so they are parallel. Line r is translated 3 units up, and line s is rotated 90° counterclockwise about the origin. Are r′ and s′ parallel?
Answer: B
The property is about moving both lines with the same motion. Here r′ is the horizontal line through (0, 3) and s′ is the vertical line through (-2, 0), so they cross. Choice A applies the property to two different motions. Choice D is false: a half turn keeps a line parallel to itself.
Question 12 of 20 · Multiple Choice
Which experiment verifies that a reflection keeps angle measures?
Answer: A
To test what a reflection does, you must reflect the angle and measure both the original and the image. Choice B never uses a reflection. Choice C measures how far the angle moved, which is not its size. Choice D measures side lengths, which do not tell you the opening of the angle.
Question 13 of 20 · Multiple Choice
Line n is the horizontal line through (0, 3). It is reflected across the x-axis. What is its image?
Answer: B
Every point of n is 3 units above the x-axis, so its image is 3 units below it: the horizontal line through (0, -3). A reflection takes a line to a line. Choice A swaps the coordinates of (0, 3). Choice C assumes the line does not move, which happens only for the x-axis itself or a line crossing it at a right angle. Choice D swaps the coordinates and changes a sign.
Question 14 of 20 · Multiple Choice
Lena measures an angle as 60° and its reflection image as 120°. What is the best explanation?
Answer: C
A reflection keeps angle measures, so both readings should be 60°. A protractor has two scales, and 180° - 60° = 120°, so one reading came from the wrong scale. Choices A, B and D all assume the reflection changed the angle, which the experiments in this lesson rule out.
Question 15 of 20 · Short Answer
Triangle UVW has vertices U(-6, -5), V(-1, -5) and W(-4, -2). Translate it 5 units right and 6 units up. List the image vertices, and show that UV = U′V′ and that angle V equals angle V′.
U′(-1, 1), V′(4, 1), W′(1, 4). UV and U′V′ are both horizontal and 5 units long. From V to W and from V′ to W′ the side goes 3 left and 3 up, corner to corner through 3 grid squares, so both angles measure 45°.
Question 16 of 20 · Short Answer
Segment CD runs from C(2, 1) to D(2, 6). Reflect it across the y-axis. Give the image endpoints, compare the lengths, and say whether C′D′ is parallel to CD.
C′(-2, 1) and D′(-2, 6). Both segments are vertical and 5 units long. Both are vertical, 4 units apart, so C′D′ is parallel to CD.
Question 17 of 20 · Short Answer
Triangle XYZ has vertices X(1, 1), Y(3, 1) and Z(1, 5). Rotate it 90° counterclockwise about the origin. List the image vertices and explain how the side lengths and the right angle compare.
X′(-1, 1), Y′(-1, 3), Z′(-5, 1). XY is horizontal and 2 units long; X′Y′ is vertical and 2 units long. XZ is vertical and 4 units long; X′Z′ is horizontal and 4 units long. The right angle at X is still a right angle at X′, and YZ and Y′Z′ measure the same with a ruler (about 4.5 units).
Question 18 of 20 · Short Answer
Line a passes through (0, 0) and (2, 5), and line b passes through (3, 0) and (5, 5). Translate both lines 4 units left and 1 unit down. Give the images of the four points and show that the image lines are parallel.
(-4, -1) and (-2, 4) on a′; (-1, -1) and (1, 4) on b′. Each image line goes 2 across and 5 up, the same steps as a and b, so a′ and b′ point the same way and are parallel.
Question 19 of 20 · Short Answer
Kai says a transformation took a quadrilateral with angles 90°, 90°, 110° and 70° to a quadrilateral with angles 90°, 90°, 100° and 80°. Could it have been a rotation, a reflection or a translation? Explain.
No. Rotations, reflections and translations keep every angle measure, so the image would need angles of 90°, 90°, 110° and 70°. The 110° and 70° angles became 100° and 80°, so the shape changed. Both sets add to 360°, so the sum alone cannot tell the difference.
Question 20 of 20 · Short Answer
A designer rotates a rectangular logo that is 12 cm wide and 5 cm tall by 90°. What are the width and height of the image, and are its corners still right angles? Explain.
The image is 5 cm wide and 12 cm tall. A rotation keeps every side length, but a quarter turn makes the 12 cm sides vertical and the 5 cm sides horizontal. It keeps angle measures, so all four corners are still 90°.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 8.G.A.1 mean?
8.G.A.1 means students check by experiment what rotations, reflections and translations do to figures. They move figures with tracing paper, transparencies or software and measure, and they find that lines stay lines, lengths and angle measures stay the same, and parallel lines stay parallel.
What are rigid motions in 8th grade math?
Rigid motions are the moves that do not change size or shape: translations (slides), reflections (flips over a line) and rotations (turns about a point). In 8.G.A.1 students discover this by measuring. In 8.G.A.2 they use sequences of these moves to define congruent figures.
What does "verify experimentally" mean in 8.G.A.1?
It means checking a fact by doing and measuring, not by proving it. Students trace a figure, move the tracing, and compare side lengths with a ruler and angles with a protractor. Formal proofs of these properties come later, in high school geometry.
Does 8.G.A.1 require coordinate rules for transformations?
No. A coordinate grid is useful for drawing images and counting lengths, but describing the effect of each move with coordinates is the focus of the next standard, 8.G.A.3. In 8.G.A.1 students can work on plain paper, on a grid, or in software.
What stays the same under a rotation, reflection or translation?
Lengths of segments, measures of angles, straightness of lines, and parallelism all stay the same. What can change is where the figure is, which way its sides point, and, for a reflection, whether the vertices go around clockwise or counterclockwise.
Why does the standard say lines go to lines?
Because it is not true for every transformation. A rule that bends the plane, like a funhouse mirror, can turn a line into a curve. Rigid motions never do: when students trace a straight segment and move the paper, it stays straight. The standard asks them to notice and check this.
Are dilations part of 8.G.A.1?
No. A dilation enlarges or shrinks a figure from a center point, so it changes lengths. Dilations appear in 8.G.A.3 and 8.G.A.4, where students compare them with rigid motions and use them to study similar figures.
What mistakes do students make with rigid motions?
Many students reflect across the wrong axis, rotate about a point other than the named center, or move a figure in the wrong direction for a translation. Some think a rotated side got longer because it now points a new way. Measuring both figures after every move catches these errors.
How does 8.G.A.1 connect to 8.G.A.2?
8.G.A.1 shows that rigid motions keep lengths and angles. 8.G.A.2 uses that fact: two figures are congruent when a sequence of rotations, reflections and translations moves one exactly onto the other, and students describe such sequences.
How does this standard prepare students for high school geometry?
In high school, students define rotations, reflections and translations precisely (HSG.CO.A.4) and use them to prove that triangles are congruent. The facts students measure in grade 8 become the starting points for those proofs.
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Related Standards
5 standards
These standards connect to 8.G.A.1: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
4.G.A.1Prerequisite
Draw points, lines, segments, rays, angles, and perpendicular and parallel lines
Lesson coming soon
4.MD.C.6Prerequisite
Measure angles in whole-number degrees with a protractor and sketch given angles
Lesson coming soon
Alongside
8.G.A.3Parallel
Describe the effect of dilations, translations, rotations and reflections using coordinates
