Defining Rigid Motions
Aligned to G-CO.A.4 — Common Core State Standards for Mathematics.
What this lesson teaches
Each rigid motion has a precise definition using basic objects. A rotation about center C by angle t sends each point P to the point P' with CP' = CP and angle PCP' = t. A reflection across line m sends each point to its mirror image, so m is the perpendicular bisector of the segment joining a point and its image. A translation slides every point the same distance in the same direction.
Worked example
Describe a reflection across line m precisely. For a point P not on m, its image P' satisfies two conditions: the segment PP' is perpendicular to m, and m cuts PP' exactly in half. So m is the perpendicular bisector of PP'. A point already on m maps to itself.
Practice questions
- In a rotation of 60° about center C, what two things stay equal for a point P and its image P'?
- A translation moves A(0,0) to (4,3). Where does it move B(2,1), and how do you know the direction and distance match?
- Explain why, under a reflection, any point on the mirror line does not move.
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Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Definir los movimientos rígidos
Cada movimiento rígido tiene una definición precisa con objetos básicos. Una rotación sobre el centro C por un ángulo t envía cada punto P al punto P' con CP' = CP y ángulo PCP' = t. Una reflexión sobre la recta m envía cada punto a su imagen especular, así que m es la mediatriz del segmento que une un punto con su imagen. Una traslación desliza cada punto la misma distancia en la misma dirección.
Ejemplo: Describe con precisión una reflexión sobre la recta m. Para un punto P fuera de m, su imagen P' cumple dos condiciones: el segmento PP' es perpendicular a m, y m corta PP' exactamente por la mitad. Así, m es la mediatriz de PP'. Un punto que ya está en m se mapea a sí mismo.
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