Math101 · Independent Ontario learning library
Math101learn.math101.caReflections
A reflection maps every point across a mirror line so that the line is the perpendicular bisector of the segment joining point and image.
Reflections model symmetry, optics, design, and congruence. Their perpendicular-bisector definition supports constructions and coordinate rules.
Intuition and core definition
A reflection maps every point across a mirror line so that the line is the perpendicular bisector of the segment joining point and image. Reflections preserve lengths and angles but reverse orientation. Points on the mirror line remain fixed.
Notation, language, and conditions
Common coordinate rules are across the $x$-axis $(x,y)\mapsto(x,-y)$, $y$-axis $(x,y)\mapsto(-x,y)$, line $y=x$: $(x,y)\mapsto(y,x)$, and vertical line $x=a$: $(x,y)\mapsto(2a-x,y)$.
Why this idea matters
A reflection reverses orientation while preserving length and angle, with the mirror line perpendicularly bisecting every point-image segment.
A dependable method
- Identify or construct the line of reflection.
- From each point, draw a perpendicular to the mirror line.
- Place the image the same distance on the opposite side.
- Apply an exact coordinate rule when the mirror is a standard line.
- Check midpoint and perpendicularity for point-image segments.
Worked example
Representations and interpretation
Folding the plane along the mirror makes corresponding points coincide. Coordinate rules encode the same equal-distance and perpendicular-bisector conditions algebraically.
Reasoning about variations
Reflecting across $y=x$ swaps coordinates, not signs. A reflection across an arbitrary line can be built with perpendicular projections or composed from rotations and standard reflections.
Common mistakes
How to check your work
- Find the midpoint of each point-image pair and confirm it lies on the mirror.
- Confirm the joining segment is perpendicular to the mirror.
- Compare corresponding side lengths and reversed orientation.
Practice
- Reflect $(5,-3)$ across the $y$-axis.
- Reflect $(-2,6)$ across $y=x$.
- What happens to a point on the reflection line?
Answers and brief solutions
Show answers
- $(-5,-3)$ Only the $x$ sign changes.
- $(6,-2)$ Swap the coordinates.
- It remains fixed Its distance to the mirror is zero.
Synthesis and transfer
To verify a reflected coordinate figure, connect each original vertex to its image; every connector should meet the mirror line at a right angle and equal half-distances.
Reflecting $P=(4,1)$ across the vertical line $x=-2$ places the image at $P'=(-8,1)$ because the midpoint's $x$-coordinate is $-2$. Segment $PP'$ is horizontal and therefore perpendicular to the mirror line, with equal six-unit distances on both sides. Points on the mirror remain fixed, providing useful anchors for an entire figure. A reflection preserves lengths and angle measures but reverses clockwise orientation. Applying the same reflection twice returns every point to its start, so the transformation is its own inverse; that property can check a coordinate rule without relying on a drawing.
Related topics
Teaching and accessibility note
Explore the idea
Vector and matrix transform
Change one quantity at a time and connect what moves to Reflections.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Reflect $(5,-3)$ across the $y$-axis.
- Only the $x$ sign changes.
End of lesson
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