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Symmetry and movements

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Here you can find out which congruence maps there are, how to recognize and construct them. "Congruence" comes from Latin and means "agreement". It is therefore a matter of images that transform figures into congruent figures. The term "symmetry" comes from the Greek (syn- = together, metron = measure) and means something like "uniformity". A figure is symmetrical if it can be mapped onto itself by a congruence map (with the exception of the identical one).

Axis mirroring

An image is called an axis mirroring if it has the following properties: A point P and its image point P ter mirrored on the axis lie on a straight line perpendicular to the axis. Both have the same distance to the axis. Points on the axis are mapped onto themselves, they are fixed points. You can imagine an axis mirroring like this: It is a movement in which figures are "flipped over" on a straight line. The straight line is called the mirror axis. You can also create the image of a figure by setting up a mirror on the mirror axis and tracing the mirrored figure. A reflection has the following properties: The original of a figure and its mirror image are congruent. Lines are mapped to lines of equal length, angles to angles of the same size, straight lines back to straight lines, whereby the parallelism of straight lines is also retained. However, the sense of rotation of a lettering on the figure changes. If the points of the original figure are labeled counterclockwise, the corresponding pixels are labeled clockwise. It is said that the original and its mirror image are oriented in opposite directions.
Axis reflections are true to length, angle and line.
In the case of an axis mirroring, the orientation is reversed.
A figure is called axially symmetrical if it has at least one mirror axis, i.e. one half of the figure is the mirror image of the other half. There are figures with several axes of symmetry. The circle even has an infinite number of axes of symmetry.
Mirror a triangle on a straight line
Reflect the triangle on the given axis.

rotation

A mapping is called a rotation about Z by the angle if it has the following properties: The image point of Z is Z, that is, Z is a fixed point. For every other point P, the following applies: P and the image point P 'have the same distance from Z. and enclose the angle with Z. The fixed point Z is called the center of rotation, the angle is called the angle of rotation. You can imagine a rotation as follows: A rotation is a movement in which a point Z remains fixed. All other points move in circles around this point at a fixed angle. A rotation has the following properties: The original of a figure and its mirror image are congruent. Lines are mapped to lines of equal length, angles to angles of the same size, lines to lines again, whereby the parallelism of lines is also retained. The direction of the lettering on the figure does not change. If the points of the original figure are labeled counterclockwise, the corresponding image points are also labeled counterclockwise. It is said that the original and its mirror image are oriented in the same way. A rotation can also be generated by two consecutive axis mirroring. To do this, choose two half-lines that include the angle of rotation. You choose one of the legs as the first mirror axis, and the bisector of the included angle of rotation as the second mirror axis.A figure is called rotationally symmetrical if it has a center of rotation, i.e. there is a point around which the figure can be rotated so that it is in line with itself. The angle of rotation must be smaller than and larger than.
Rotations are true to length, angle and straight line and maintain the orientation
Flip the triangle.

Point mirroring

An image is called a point reflection at point Z if it has the following properties: The image point of Z is Z. For every other point P the following applies: P and the image point P 'are at the same distance from Z, and Z is the midpoint of the segment. Each point reflection is a rotation around Z with the angle of rotation. You can imagine the point reflection as a rotation around Z with the angle of rotation. The point reflection has the following properties: The original of a figure and its mirror image are congruent. Angle to equal angles, straight lines back to straight lines, whereby the parallelism of straight lines is also retained. The direction of the lettering on the figure does not change. If the points of the original figure are labeled counterclockwise, the corresponding image points are also labeled counterclockwise. It is said that the original and its mirror image are oriented in the same way. A point mirroring can also be created by two consecutive axis mirroring. The two mirror axes must go through Z and be perpendicular to each other. A figure is called point-symmetrical if it has a point of symmetry, i.e. there is a point around which the figure can be turned around without changing it.
Point reflections are true to length, angle and line and maintain orientation.
A point reflection is a rotation around.
Mirror a triangle at a point
Reflect the triangle at the given point Z.

shift

A mapping is called a shift (also parallel shift) if it has the following properties: Each point of a figure is shifted by the same length and in the same direction. The shift of a point P to its image point P "is indicated by a shift arrow. A figure and the shifted figure are congruent, because every shift is a congruence mapping: Lines are mapped to lines of equal length, angles to angles of the same size, straight lines back to straight lines, The parallelism of straight lines is also retained. The direction of the lettering does not change. The original figure and the shifted figure are oriented in the same way. A parallel shift can also be achieved by two consecutive axis mirroring. You draw a line with its length and direction through the As the first mirror axis you choose the straight line that is perpendicular to one of the endpoints of this line, and the second mirror axis is the mid-line perpendicular to the line.
Shifts are true to length, angle and straight line and maintain the orientation.
Shifts can also be represented by two parallel axis reflections carried out one behind the other. The distance between the two axes is half the length of the shift.
Move the triangle along the given arrow.