Geometry transformations describe how a figure moves on a plane. In congruence proofs, the most important transformations are rigid motions because they preserve side lengths and angle measures. If one figure can be moved exactly onto another by rigid motions, the two figures are congruent.
This idea connects visual geometry on a coordinate grid to formal proof.
Key Facts
- Rigid motions preserve distance and angle measure.
- If a sequence of rigid motions maps Figure A onto Figure B, then Figure A ≅ Figure B.
- Translation rule: (x, y) -> (x + a, y + b).
- Reflection over the x-axis: (x, y) -> (x, -y).
- Reflection over the y-axis: (x, y) -> (-x, y).
- Rotation 90° counterclockwise about the origin: (x, y) -> (-y, x).
Vocabulary
- Transformation
- A transformation is a rule that moves or changes points of a figure on a plane.
- Rigid motion
- A rigid motion is a transformation that preserves distances and angle measures.
- Translation
- A translation slides every point of a figure the same distance in the same direction.
- Reflection
- A reflection flips a figure across a line so that corresponding points are the same distance from the line.
- Congruent figures
- Congruent figures have the same size and shape, so all corresponding sides and angles match.
Common Mistakes to Avoid
- Using a dilation in a congruence proof is wrong because dilation changes size unless the scale factor is 1.
- Matching vertices in the wrong order is wrong because corresponding sides and angles must pair correctly for a valid congruence statement.
- Calling every slide a translation without checking direction and distance is wrong because each point must move by the same vector.
- Forgetting that reflections reverse orientation is wrong because a reflected figure has the opposite clockwise or counterclockwise order of vertices.
Practice Questions
- 1 Triangle A has vertices (1, 2), (4, 2), and (2, 5). Translate it by the rule (x, y) -> (x + 3, y - 4). What are the coordinates of the image?
- 2 Quadrilateral P has vertices (-2, 1), (-1, 4), (2, 3), and (1, 0). Reflect it over the y-axis. What are the coordinates of the image?
- 3 A polygon is reflected over the x-axis and then translated 5 units right. Explain why the final image is congruent to the original polygon.