Geometry Unit 4
The dilation is a reduction if the scale factor is between
0 and 1.
Translations
A translation is a transformation in which a figure is simply moved from one location to another. This is also known as a slide. In a translation, every point in the figure moves the same distance and in the same direction. Sometimes a slide arrow is used to show the distance and direction of the movement. A transformation changes only the location of the figure while preserving orientation and size.
Take Note Theorem 9-1
A translation or rotation is a composition of two reflections.
Reflections
Another type of transformation is a reflection. The reflection you see in a mirror is a reverse image of you. A reflection in geometry flips a figure over a line. A reflection is also known as a flip. The line that the figure is reflected over is called the line of reflection. Each point in the image is the same distance from the line of reflection as the original point. A reflection changes the direction of a figure. That is, the figure under a reflection will appear to be a backward version of the original. The line of reflection can be vertical, horizontal, or diagonal.
How can you identify symmetry in a 3-D object?
For reflectional symmetry, try to visualize a plane slicing the object. For rotational symmetry, try to imagine rotating the object about a line.
Take Note Theorem 9-3: Fundamental Theorem of Isometries
In a plane, one of two congruent figures can be mapped onto the other by a composition of at most three reflections.
How do you identify rotational symmetry?
Look for a possible center point. Think about the angles formed by joining preimage-image pairs to the center. All these angles must be congruent for the figure to have rotational symmetry.
One type of transformation is a translation, which happens when
an object simply changes its position.
An angle of rotation is twice the
angle formed by the lines of reflection.
In geometry, when a figure is enlarged or reduced so that the image is similar to the preimage, a type of transformation called _________occurs
dilation
There are only _____ isometries.
four
A dilation is an enlargement if the scale factor is
greater than 1.
The resulting figure is the
image
However, unlike translation, reflection, and symmetry, dilations are not
isometric
Since the size and shape do not change, a translation is also
isometric
Translation, rotation, reflection, and glide reflection are
isometries. Dilation is not an isometry.
A figure with 180° 180 degrees rotational symmetry also has
point symmetry. Each segment joining a point and its 180° 180 degrees rotation image passes through the center of rotation. A square, which has both 90° 90 degrees and 180° 180 degrees rotational symmetry, also has point symmetry.
A transformation of a geometric figure is a change in the
position, shape, or size of the figure.
Drawing a diagram of a situation before performing the transformation can help
predict where the image will be.
In a transformation, the original figure is the
preimage
Rotations can be written using
reflections
A transformation that preserves distance and angle measures is called a
rigid motion.
two reflections across intersecting lines
rotation
If a rigid motion results in a figure being mapped onto itself, then the figure is considered to have
symmetry. Both plane figures and three-dimensional objects can have symmetry.
A glide reflection is
the composition of a translation (a glide) and a reflection across a line parallel to the direction of translation.
An isometry is a transformation in which
the preimage and image are congruent.
A figure has line symmetry or reflectional symmetry if
there is a reflection for which the figure is its own image. The line of reflection is called a line of symmetry. It divides the figure into congruent halves.
A figure has rotational symmetry if
there is a rotation of 180° 180 degrees or less for which the figure is its own image. The angle of rotation for rotational symmetry is the smallest angle needed for the figure to rotate onto itself.
Reflections are another form of
transformation and are isometric.
an isometry is a
transformation in which preimage and image are congruent. Another term to describe this is rigid motion.
A translation is a
transformation that maps all points of a figure the same distance in the same direction.
A translation is a
transformation that maps all points of a figure the same distance in the same direction. A translation is an isometry.
When classifying isometries, using the process of elimination can help determine
which isometry is used.
Rotations
Rotating something means to turn it. In geometry, a rotation is a transformation that turns a figure around a fixed point called the center of rotation. A rotation is also known as a turn. A rotation is described by the direction (clockwise or counterclockwise) the figure is turned and by the angle through which the figure is rotated around the center of rotation. The center of rotation can be a point on the figure or outside of the figure. A rotation changes the orientation of a figure. That is, the image will appear to be on its side or upside down.
A transformation is
a change of the position, orientation, direction, or size of a figure. The result of a transformation is the image of the original.
A composition of transformations is
a combination of two or more transformations. In a composition, you perform each transformation on the image of the preceding transformation. In general, the composition of any two translations is another translation.
