What is the reflection?
In geometry, reflection is a type of transformation that creates a mirror image of the original figure. The shape is mirrored about a line known as the line of reflection. When a figure is said to be a reflection of another figure, each point in that figure and each corresponding point in the reflected figure are equidistant from the line of reflection.
A real-life example is the reflection of Mount Hood in Mirror Lake. The image you see of your face in the mirror is a reflection of your face.
Reflection in coordinates plane:
A reflection transformation could be applied to both the X and Y axes.
Reflection in X-axis:
The X-coordinate of a point remains the same when it is reflected across the X-axis. The Y-coordinate, on the other hand, is changed to the opposite sign.
As a result, the point (x,y) is reflected across the X-axis as (x,-y).
Reflection in Y-axis:
The Y-coordinates of a point remain unchanged when reflected across the Y-axis. However, the signs of the X-coordinates are reversed.
As a result, the point (x, y) is reflected along the Y-axis (-x, y).
Solved Examples on Reflection:
Example 1:
Determine whether the green figure is a reflection of the blue figure.
(a)
Solution:
The blue figure can be flipped to form the green figure. The points of the blue figure and corresponding points of the green figure are equidistant from the line of reflection as well.
So, the green figure is a reflection of the blue figure.
(b)
If the green figure were flipped, it would point to the left. As we can see from that, the green figure is not a reflection of the blue figure.
Example 2:
A (2, 2), B (4, 2), C (4, 4), D (3, 5), and E (2, 4) are the vertices of a pentagon. Draw the figure and its reflection in the x-axis and the y-axis, respectively. What are the coordinates of the image?
Solution:
1: Reflection about x-axis
Points A and B are 2 units above the x-axis so plot A’ and B’ will 2 units below the x-axis. Points C and E are 4 units above the x-axis so plot C’ and E’ are 4 units below the x-axis. Point D is 5 units above the x-axis so, plot D’ will be 5 units below the x-axis.
For reflection in the x-axis, the points of the figure and the corresponding points of the reflected image are aligned vertically.
The coordinates of the image are A’(2, -2), B’ (4, -2), C’ (4, -4), D’(3, -5) and E’(2, -4).
1: Reflection about y-axis
Points A and E are 2 units to the right of the y-axis, so plot A’ and E’ are 2 units to the left of the y-axis and points B and C are 4 units to the right of the y-axis so plot B’ and C’ are 4 units to the left of the y-axis. Point D is 3 units to the right of the y-axis so point D’ is 3 units to the left of the y-axis.
For a reflection in the y-axis, the points of the figure and the corresponding points of the reflected image are aligned horizontally.
The coordinates of the image are A’(-2, 2), B’ (-4, 2), C’ (-4, 4), D’(–3, 5) and E’(-2, 4).
Example 3:
A hexagon with vertices A(0, 0), B(-2, 0), C(-1, 3), D(-4, 3), E(-4, 3), and F(-4, 3) is used by a designer to create a shirt. To create the design, the designer reflects the hexagon in the X-axis. Find the reflected image’s coordinates. Then, in the coordinate plane, draw the design.
Solution:
The hexagon is reflected in the x-axis. For the image, take the opposite sign of each y-coordinate of the vertices. There is no change in the x-coordinates.
(x, y) ⟶ (x, -y)
A (2, 0) ⟶ A’(2, 0)
B (5, 0) ⟶ B’(5, 0)
C (6, 2) ⟶ C’(6, -2)
D (5, 3) ⟶ D’(5, -3)
E (2, 3) ⟶ E’(2, -3)
F (1, 2) ⟶ F’(1,- 2)
Hence, the coordinates of reflected images are A’(2, 0), B’(5, 0), C’(6, -2), D’(5, -3), E’(2, -3) and F’1, -2.
Example 4:
In terms of reflections, describe the relationship between the given point and point A(7,4).
a. B(7, -4)
b. C(-7, 4)
Solution:
- On comparing A(7, 4) with B(7, -4), it is observed that the x-coordinate is the same but the signs of the y coordinate are opposite. Hence, the point (x, y) is reflected across the x axis (x, -y).
- Hence, the reflection of the point A(7, 4) across the x-axis is B7, -4.
- On comparing A(7, 4) with C(-7, 4), it is observed that the y-coordinate is the same but the signs of the x coordinate are opposite. Hence, point (x, y) is reflected across the y axis (-x, y).
If the coordinates of the shape before reflection are 1,3,5 then your reflected image’s coordinates are 1′,3′,5′ respectively.
Reflection about an axis means that the reflection takes place keeping that axis as the reference. For example, if a point or a figure is reflected about the x-axis it means that the x-axis will act as the line of reflection.
When we reflect a point about the origin both the signs of the x and y coordinates of the image will be the opposite of the coordinates of the given point.
If a point A (x,y) is reflected in the origin,
(x,y)⟶(-x,-y)
The coordinates of the reflected image is A’ (-x,-y)