A reflection is a transformation that flips a figure across a mirror line on the plane. The original figure and its reflected image have the same size and shape, but they face opposite directions. Reflections are important because they connect geometry, symmetry, coordinate rules, and real world mirror images.
On a coordinate plane, reflections can be described exactly using ordered pairs.
Key Facts
- Reflection over the x-axis: (x, y) -> (x, -y).
- Reflection over the y-axis: (x, y) -> (-x, y).
- Reflection over the line y = x: (x, y) -> (y, x).
- Reflection over the line y = -x: (x, y) -> (-y, -x).
- A reflection preserves distance, angle measure, side length, area, and perimeter.
- The mirror line is the perpendicular bisector of each segment joining a point to its reflected image.
Vocabulary
- Reflection
- A reflection is a transformation that flips every point of a figure across a line to create a mirror image.
- Line of reflection
- The line of reflection is the mirror line that each point and its image are equally far from.
- Image
- The image is the new figure produced after a transformation is applied to the original figure.
- Preimage
- The preimage is the original figure before a transformation is applied.
- Isometry
- An isometry is a transformation that preserves distances and angle measures.
Common Mistakes to Avoid
- Changing both coordinates when reflecting over one axis is wrong because a reflection over the x-axis only changes the sign of y, while a reflection over the y-axis only changes the sign of x.
- Using the rule for y = x when reflecting over y = -x is wrong because y = x swaps the coordinates, but y = -x swaps the coordinates and changes both signs.
- Assuming the reflected figure has a different size is wrong because reflections are isometries, so side lengths, angles, area, and perimeter stay the same.
- Drawing the image at an unequal distance from the mirror line is wrong because each point and its image must be the same perpendicular distance from the line of reflection.
Practice Questions
- 1 Triangle ABC has A(2, 3), B(5, 1), and C(4, 6). Find the coordinates of A'B'C' after reflection over the x-axis.
- 2 Point P(-3, 7) is reflected over the line y = x, then the result is reflected over the y-axis. What are the final coordinates?
- 3 A triangle is reflected across a vertical line. Explain how its orientation changes and which properties of the triangle remain unchanged.