Lesson 2 — Reflections
Big idea: A reflection flips a figure over a line; the line is the perpendicular bisector of every segment joining a point to its image.
Words to know
| Word | Meaning |
|---|---|
| Reflection of a point through a line | The image point such that the line is the perpendicular bisector of the segment joining the point and its image. |
| Composite | A transformation made by applying two or more transformations in order. |
| Translation | A slide; can be made by two reflections over parallel lines. |
| Rotation | A turn; can be made by two reflections over intersecting lines. |
| Magnitude | The distance of a translation or the angle of a rotation. |
Concepts to know
- A reflection is an isometry: distances and angles are preserved.
- The reflecting line is the perpendicular bisector of the segment from a point to its image.
- A point on the reflecting line stays fixed.
- Two reflections over parallel lines produce a translation.
- The translation distance is twice the distance between the lines.
- Two reflections over intersecting lines produce a rotation.
- The rotation angle is twice the angle between the lines.
- Construction 8: Reflect a point through a line.
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Reflect a point | Draw a perpendicular from the point to the line; mark the same distance on the other side. |
| Reflect a figure | Reflect each vertex, then connect the images. |
| Find the reflecting line | It is the perpendicular bisector of the segment joining a point and its image. |
| Identify a composite | Two parallel reflections → translation; two intersecting reflections → rotation. |
| Find the magnitude | Translation: twice the line distance. Rotation: twice the angle between lines. |
| Check a reflection | The line must bisect the segment at 90°. |
Worked example
Reflect point P over line ℓ.
- Draw a perpendicular from P to ℓ; label the intersection M.
- Extend the perpendicular past M.
- Mark P′ so that PM = MP′.
Then ℓ is the perpendicular bisector of PP′.
Watch out
- The reflecting line is the perpendicular bisector, not just any line through the midpoint.
- A reflection reverses orientation.
- Two reflections over parallel lines make a translation, not a rotation.
- Two reflections over intersecting lines make a rotation, not a translation.