Track translations, reflections and rotations using coordinates and distance preservation.
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On a narrow screen, scroll diagrams horizontally to read every label. Keyboard users can focus a diagram and use the arrow keys.
Try
Translate the triangle (0,0),(2,0),(0,1) by (4,1). Its image is (4,1),(6,1),(4,2). The same vector moves every vertex. Adding different vectors to different points would generally distort the triangle.
Predict
Predict which properties survive a rigid motion. Distances and angles do; a reflection reverses orientation while translations and rotations preserve it. A rotation must include its center, angle and direction. A reflection must name its mirror line. The word “turn” alone is insufficient.
Experiment
Follow the dashed displacements in the diagram. Compare each original side with its image and calculate one squared distance in both positions. Then reflect a hand-drawn copy across a line: clockwise vertex order becomes counterclockwise, even though side lengths remain unchanged.
A triangle and its translation by (4,1), using equal axis scales. Every vertex receives the same displacement; the two triangles have equal shape and size.
Understand
Worked argument: reflection across the x-axis sends (x,y) to (x,−y). For two points, the new squared distance is (x₂−x₁)²+(−y₂+y₁)², equal to the original because squaring removes the sign of the vertical difference. Thus distances are preserved. A 90° counterclockwise rotation about the origin sends (x,y) to (−y,x); substituting into the same distance formula also preserves the squared sum.
Invalid proof: “Every coordinate changed, so the size changed.” Coordinates depend on position and axes. Congruence concerns distances and angles. Conversely, merely preserving one side length does not show a transformation preserves the whole figure.
Apply
State each motion fully, calculate images in the requested order, and then compare. In written proofs, arbitrary coordinates establish preservation generally; a single matching pair only checks one example. Keep transformations separate from changing coordinate labels without moving the figure.
Continue
Depth: any composition of rigid motions is rigid because each step preserves distances. Order can still matter: a translation followed by a rotation usually differs from the reverse order. Investigate the special case of two translations, whose vectors add in either order.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
Translate (−3,5) by vector (7,−2). What is the image x coordinate?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Rotate (4,−7) by 90° counterclockwise about the origin. What is the image y coordinate?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
What is the reflection of (3,−2) across the y-axis?
Which motion reverses the clockwise order of a noncollinear triangle's vertices?
Prove that translating every point by (u,v) preserves the distance between arbitrary points (x1,y1) and (x2,y2).
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Show by an example that translating by (1,0) and rotating 90° counterclockwise about the origin give different results in different orders.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
