We will study reflection, the last type of figure transformation. It is a transformation that creates a figure as if it were reflected in a mirror. Finally, we will summarize that combining the three types of transformations can produce a figure that overlaps the original figure, meaning that the figures are congruent.
What Is a Reflection?
Using a line ℓ as the axis of symmetry, moving a figure to the opposite side as if it were reflected in a mirror is called a reflection (line reflection).
Properties of a reflection:
・The segment connecting corresponding points is perpendicularly bisected by the axis of symmetry ℓ.
・Points on the axis of symmetry do not move.
・Corresponding sides have equal lengths, and corresponding angles are equal.
Point A was reflected across line ℓ to create point A'.
(1) What is the relationship between segment AA' and line ℓ?
(2) If M is the point where AA' and ℓ intersect, what is the relationship between AM and MA'?
Solution:
(1) AA' ⊥ ℓ (perpendicular)
(2) AM = MA' (M is the midpoint of AA')
In other words, ℓ is the perpendicular bisector of AA'.
Reflections in Coordinates
- Reflection across the x-axis: (x, y) → (x, −y) (the sign of the y-coordinate is reversed).
- Reflection across the y-axis: (x, y) → (−x, y) (the sign of the x-coordinate is reversed).
- Reflection across the origin (the same as a 180° rotation): (x, y) → (−x, −y).
Find the coordinates of the point obtained by reflecting P(3, −5) across each of the following lines.
(1) x-axis (2) y-axis
Solution:
(1) Reflection across the x-axis: (x, y) → (x, −y), so (3, 5)
(2) Reflection across the y-axis: (x, y) → (−x, y), so (−3, −5)
Congruent Figures
When two figures can be made to overlap by combining translations, rotations, and reflections, the figures are called congruent.
For two congruent figures, corresponding sides have equal lengths and corresponding angles have equal measures.
- Reflection: reflecting a figure across one line, the axis of symmetry.
- The segment connecting corresponding points is perpendicularly bisected by the axis.
- Reflection across the x-axis: (x, y)→(x,−y); reflection across the y-axis: (x, y)→(−x, y).
- Congruent figures: figures that can be made to overlap through the three types of transformations.
Practice Problems
- Find the coordinates of the point obtained by reflecting A(−2, 4) across the x-axis.
- Find the coordinates of the point obtained by reflecting B(5, −3) across the y-axis.
- Select all the correct statements about reflections.
ア) The orientation of the figure does not change. イ) Points on the axis of symmetry do not move. ウ) Corresponding sides have equal lengths. エ) The segment connecting corresponding points is parallel to the axis.
Answers and Explanations
- Answer: (−2, −4)
Explanation: Reflection across the x-axis reverses the sign of the y-coordinate. - Answer: (−5, −3)
Explanation: Reflection across the y-axis reverses the sign of the x-coordinate. - Answer: イ、ウ
Explanation: A reflection reverses the orientation of the figure, and the segment connecting corresponding points is perpendicular to the axis. Points on the axis do not move, and corresponding sides have equal lengths.