Week 11 Flashcards
Similar shapes
Shapes that are related by a dilation, possibly along with a rigid motion of some sort
Two shapes are similar if
the points in the two shapes can be matched so that every pair of corresponding angles are of the same size and the ratios from every pair of corresponding lengths all equal the same value, called the scale factor
Scale factor
The common ratio of the image length to the original length in similar shapes
Dilation
A matching of the points of a plane (or space) such that the size of every angle is the same as in its image and such that the ratio of an image length to the original length is always the same value, called the scale factor
Center of a dilation
The point with which the matching is done
Image
The corresponding point or shape that a transformation gives for the point or shape
Original
Shape before the transformation
Comparison language
Language that communicates additive comparisons, multiplicative comparisons, or both
Similar triangles
Vertices can be matched so that two pairs of corresponding angles are of the same size
Ratio of areas
Square of the scale factor, ie. scale of perimeter is 4, scale of area is 16
Ratio of volumes
scale factor cubed
Transformation geometry
A term for a school geometry topic that features rigid motions and size changes
Rigid motion or isometry
A movement that does not change lengths and angle sizes
Translation
A type of rigid motion in which the image of a point is the point that is a fixed distance in a fixed direction from the original point, ie. moon across the sky
Rotation about a point
A 2D rigid motion in which the plane is turned about a point called the center of the rotation, ie. flower petals
Reflection in a line
2D rigid motion in which a point on either side of the line has its image as though the line were a mirror, ie. reflection of a mountain in water
Glide-reflection
A type of rigid motion, the composition of a translation and a reflection in a line parallel to the vector of the translation, ie. leaves alternating along a stem
Orientation
A clock direction assigned to a two-dimensional shape
Congruent shapes
Shapes for which some rigid motion gives one shape as the image of the other; this rigid motion assures that the shapes are exactly the same shape and the same size
Vector
The direction and distance associated with a given translation, often show by an arrow with the correct length and pointed in the correct direction
What tools do you need for a reflection in a line
Line
What tools are needed for a rotation with center point
Center point, angle, and clock direction
What tools are needed for a translation
Distance and direction, in form of vector (or not)
Fixed point
A point that is its own image
Composition of rigid motions
The rigid motion that describes the net effect from original shape to final shape when one rigid motion is followed by another
Composition of rigid motions formula
(second motion) o (first motion) = (the composition of the motions)
o symbol read as “after”
Glide-reflection
A type of rigid motion, the composition of a translation and a reflection in a line parallel to the vector of the translation
Any composition of rigid motions can be describe by
a single reflection, rotation, translation, or glide-reflection
Which isometries reverse orientation?
Reflection and glide-reflection
Which transformations preserve orientation?
Translation, rotation, and dilation
Does order matter in a composition of rigid motions?
Yes, sometimes
Does order always matter in a composition of rigid motions?
No, not always
9 cm is 3 times longer than 3 cm T/F
False
12 cm is 3 times longer than 3 cm T/F
True
9 cm is 3 times as long as 3 cm T/F
True
12 cm is 3 times as long as 3 cm T/F
False
Plane tessellation
A pattern made up of one or more shapes, that completely covers the plane without any gaps or overlaps, and extends infinitely in the plane in every direction
Tessellation Method 1
Arranging congruent triangles - two congruent scalene triangles can be arranged in 6 ways
Tessellation Method 2
Quadrilaterals - any quadrilateral can be used by itself to tessellate the plane, because the interior angles always sum to 360
Tessellation Method 3
Combinations of regular polygons
Which regular polygons can tessellate the plane by themselves?
Equilateral triangles, squares, and regular hexagons
Frieze
An infinite strip of a repeating pattern
A frieze is another name for a tessellation T/F
False
What is the difference between a frieze and a tessellation?
A frieze is confined to a strip and goes infinitely in one direction, a tessellation continues infinitely in all directions
Conway’s 7 types of friezes
Hop, step, sidle, spinning hop, spinning sidle, jump, spinning jump
Frieze classification - Hop is a
Translation only
Frieze classification - Step is a
Translation and glide-reflection
Frieze classification - Sidle is a
Translation and vertical reflection
Frieze classification - Spinning hop is a
Translation and rotation of 1/2 turn
Frieze classification - Spinning sidle is a
Translation, glide-reflection, and rotation
Frieze classification - Jump is a
Translation and horizontal reflection
Frieze classification - Spinning jump is a
Translation, vertical and horizontal reflection, and rotation
A:a = C:kc T/F where A is area of face of shape X, a is non-adjacent edge, C is area of face of shape Y and c is non-adjacent edge. Y is similar to X by scale factor k.
True
