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Axial symmetry in your environment
An image is characterised by axial symmetry if it can be folded in half along a line so that, when a mirror is placed along that line, the reflection and the visible half together create an image that looks exactly like the original. This line is called the mirror line or line of symmetry \latex{ (l) }.
Find a picture that is characterised by axial symmetry and fold it in half. With your compass, poke some holes through the folded image.
Unfold the image.
You can see that each point has a reflection, which is called the mirror image of the point.
For example, points \latex{ A - A’, B - B’, C - C’ }, and \latex{ D - D’ } are mirror images of each other. These pairs of points are the same distance from the axis (e.g., \latex{ D’L = DL }).
\latex{ A }
\latex{ B }
\latex{ D’ }
\latex{ D }
\latex{ B’ }
\latex{ A’ }
\latex{ l }
\latex{ L }
\latex{ C = C’ }
Examples of axial symmetry in your environment:
Example 1
The following images show parts of two drawings and their lines of symmetry. Complete the drawings.
\latex{ l }
\latex{ l }
Fold a piece of paper in half and draw half a pine tree on one side. Cut it out, then unfold the paper.
Solution
You can complete the images by drawing the mirror images of the existing parts on the corresponding half-planes.
\latex{ l }
\latex{ l }
The mirror images of the red lines and points are the blue lines and points, while the mirror images of the blue lines and points are the red lines and points.
Every point of a plane has a corresponding mirror image on the other half-plane.
The letter \latex{ A } is characterised by axial symmetry as it has a vertical line of symmetry \latex{ (l) } that divides the letter into two equal and symmetrical parts. Construct the letter \latex{ A } as shown in the image. Mark the points \latex{ A, B, C, } and \latex{ D } on it. Find the mirror images of the points. What can you observe?
\latex{ A = A’}
\latex{ C }
\latex{ C’ }
\latex{ D }
\latex{ D’ }
\latex{ B = B’ }
\latex{ l }
Points \latex{ A } and \latex{ B } are on the axis; thus, their reflections are themselves (\latex{ A = A’, B = B’ }).
The mirror image of point \latex{ C } is \latex{ C’ }, and the mirror image of point \latex{ C’ } is \latex{ C }.
The mirror image of point \latex{ D } is \latex{ D’ }, and the mirror image of point \latex{ D’ } is \latex{ D }.
The mirror image of point \latex{ C } is \latex{ C’ }, and the mirror image of point \latex{ C’ } is \latex{ C }.
The mirror image of point \latex{ D } is \latex{ D’ }, and the mirror image of point \latex{ D’ } is \latex{ D }.
Pairs of points are at equal distances from the axis, as their reflected images are on opposite sides.
The line segment connecting the point and its mirror image is perpendicular to the axis.
The line segment connecting the point and its mirror image is perpendicular to the axis.
\latex{ CB = C’B’ } and \latex{ CC’ ⊥l }
\latex{ l }
\latex{ A = A’ }
\latex{ C }
\latex{ C’}
\latex{ B }
\latex{ CB = C’B }
\latex{ A = A’ }
\latex{ C }
\latex{ C’ }
\latex{ D }
\latex{ D’ }
\latex{ l }
Points \latex{ A, C, } and \latex{ D } are on a line segment; thus, their mirror images (\latex{ A’, C’ }and \latex{ D’ }) are also on a line segment.
The line segment and its mirror image are equal in length; for example, \latex{ CD = C’D’ }.
\latex{ A = A’ }
\latex{ C }
\latex{ C’}
\latex{ D }
\latex{ D’ }
\latex{ l }
\latex{\gamma}
\latex{\gamma}'
\latex{\alpha}
\latex{\alpha}'
An angle and its mirror image are the same size; for example \latex{\gamma = \gamma ’, \alpha = \alpha ’}.
Exercises
{{exercise_number}}. Which of the following numbers and letters are characterised by axial symmetry?
Z
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ 7 }
\latex{ 8 }
\latex{ 9 }
M
L
K
J
I
H
G
F
E
D
C
B
A
N
O
P
Q
R
S
T
U
V
W
X
Y
In the following three exercises, pay attention to the colour of the figures as well.
{{exercise_number}}. There are six mistakes in the reflected image. Can you find them all?
{{exercise_number}}. Which of these drawings are characterised by axial symmetry? How many lines of symmetry do they have?
{{exercise_number}}. You can see the flags of four European countries below. Which of them has a line of symmetry? Do any of them have more than one line of symmetry?
{{exercise_number}}. The word MOM has a line of symmetry. Find similar words.
{{exercise_number}}. The following images show parts of drawings that are characterised by axial symmetry. The lines of symmetry are also shown. Complete the drawings.
\latex{ l }
\latex{ l }
\latex{ l }
\latex{ l }
{{exercise_number}}. Come up with car logos that have lines of symmetry. Draw them.
Quiz
What time is it if the clock running backwards looks like this?
