Transformations & congruence
The idea
Slide a shape across the floor, spin it, or flip it over like a pancake — it remains the same shape at the same size. These three moves are the rigid motions: translations slide, rotations turn, and reflections mirror. On the coordinate plane you already know, each becomes a precise rule for the coordinates; a translation 3 left and 2 up sends every point (x, y) to (x − 3, y + 2). Two figures are congruent exactly when some sequence of rigid motions carries one onto the other.
What makes rigid motions powerful is what they refuse to change: every distance between points and every angle survives untouched, so congruence is guaranteed rather than checked side by side. The misconception to clear is that moving a figure somehow alters it — a rotated rectangle may look tilted and unfamiliar, but its sides and angles are identical to the original's. Only a dilation, which enlarges or shrinks, changes size, and that is precisely why dilation is excluded from the rigid-motion club.
Worked example
Triangle ABC has vertices A(1, 2), B(4, 2), and C(4, 6). It is translated 3 units left and 2 units up. Find the image vertices and explain why the image is congruent to the original.
- Write the translation as a coordinate rule: 3 left subtracts 3 from x and 2 up adds 2 to y, so (x, y) → (x − 3, y + 2).
- Apply the rule to each vertex: A(1, 2) → A'(−2, 4), B(4, 2) → B'(1, 4), and C(4, 6) → C'(1, 8).
- Test a side length: AB runs from x = 1 to x = 4 at the same height, length 3, while A'B' runs from x = −2 to x = 1, also length 3.
- Test another: BC rises from y = 2 to y = 6, length 4, and B'C' rises from y = 4 to y = 8, also length 4 — distances are preserved.
- Conclude congruence: a translation moves every point the same distance in the same direction, so all sides and angles are unchanged and triangle A'B'C' is congruent to triangle ABC.
Answer. The image vertices are A'(−2, 4), B'(1, 4), and C'(1, 8), and the image is congruent because translations preserve all lengths and angles.
Check your understanding
- Why do translations, rotations, and reflections preserve distances while a dilation does not?
- How could you decide which rigid motion, or combination of motions, maps one given figure onto another?
- What stays the same and what changes about a figure's coordinates under a reflection across the x-axis?
- Why is defining congruence through motions more powerful than just comparing side lengths one by one?
Build the foundations first
Transformations & congruence builds on these concepts. If any feel shaky, start there.