Unit 1 Notes Flashcards
Mapping
same thing as Function. A correspondence of A to B IF AND ONLY IF each member of A corresponds to one and only one member of B
Transformation
When you have the same number of points in the pre-image and image. One to one function.
Isometric transformation (Rigid motion)
A transformation that preserves the distances and/or angles between the pre-image and image
Non-isometric transformation (Non-rigid motion)
A transformation that does not preserve the distance and angles between the pre-image and image
Stretch
Non-isometric transformation
Where one dimension’s scale factor is different than the dimension’s scale factor. A strecth definitely distorts the shape
Dialation
Where both dimension’s scale factors are the same. The shape is proportional, not identical. Dialation changes the size of the shape making it a NON-ISOMETRIC transformation.
what does to carry a shape onto itself mean
That the shape has symmetry
Three types of symmetries
Line symmetry, rotation symmetry, and point symmetry
Line of symmetry definition
A figure in the plane has a line of symmetry if the figure can be mapped onto itself by a reflection in the line
How many lines of symmetry do rectangles have
2
Why is the maximum lines of symmetry that a polygon can have equal to it’s number of sides
Because all sides and all angles are congruent
Rotational symmetry
A geometric figure has rotational symmetry if the figure is the image of itself under a rotation about a point through any angles who measure is strictly between 0 and 360. 0 and 360 are excluded from counting as having rotational symmetry because it represents the starting position. You can tell if a figure has at least an order of 2 if you flip the image upside down and it looks the same
Angle of rotation
The smallest angle through which the figure can be rotated to coincide with itself. This number will always be a factor of 360. To find divide 360 by the order of rotation
Order of rotation symmetry
The number of positions in which the object looks exactly the same. When determining order, the last rotation returns the object to its original position. Order 1 implies no true rotational symmetry since a full 360 degree rotation was needed.
Point symmetry
Exists when a figure is built around a point such that every point in the figure has a matching point that is:
the same distance from the central point
But in the opposite direction.
- Turn the figure upside down and see if it looks the same to see if it has point symmetru.
- All have an order of 2 and even orders
Reflection
A reflection over a line is an isometric transformation in which each point on the original figure has an image that is the same distance from the line of reflection has the original point but in on the opposite side of the line.
- The image does not change size or shape
- AA’ ll BB’ ll CC’
AA’ is not congruent to BB’ or CC’
Isometric properties of a reflection
The following stay the same
-Distance (lengths of the segments are the same)
- Angle measure (angles stay the same)
-Parallelism (things that were parallel are still parallel)
- Collinearity (points on a line, remain on the line)
After a reflection, the pre-image and image are identical
Transformation properties of a reflection
The following change:
- Distance: point son the plane move different distances, depending on their distance from the line of reflection. Points further away from the line of reflection move a greater distance than those closer to the line of reflection
- Orientation: is reversed (orientation is the order of the points about the shape)
- Special points: Points on the line of reflection do not move at all under the reflection (A + A’).
Rotation
An isometric transformation that turns a figure about a fixed point called the center of rotation. (A can = A’ if the point is on the center of rotation). AA’ is not ll to BB’
and AA’ is not congruent to BB”
Angle of rotation
Rays drawn from the center of rotation to a point and it’s image
Rotation direction
Counter clockwise- positive
Clockwise- negative
Co-terminal angles
Because angle are formed along an arc of a circle there are two ways to get to the same location, a positive direction and a negative direction.
The co-terminal angle formula finds the other positive or negative angle
Coterminal angle= initial angle+ 360 n and n is an integer like 1 or -1
Isometric properties of a rotation
The following stay the same:
-Distance (lengths of segments are the same)
-Angle measure (angles stay the same)
-Parallelism (things that were parallel are still parallel)
-Collinearity (points on a line, remain on the line)
After a rotation, the pre-image and image are identical
Transformation properties of a rotation
The following change
- distance: points in the plane move different distances, depending on their distance from the center of rotation. A point father away from the center of rotation maps a greater distance than those points closer to the center of rotation
- Orientation is the same
- Special points: the center of rotation is the only point in the plane that is unchanged
Translation
Slides an object a fixed distance in a given direction. A vector is used to describe the fixed distance and the given direction.
AA” ll BB” (a fixed idrection)
AA” is congruent to BB” (a fixed distance)
Isometric properties of a translation
- Distance
- Angle measure
- Parallelism
- Collinearity (point on a line, remain on a line)
Transformation properties of a translation
- Distances are the same: points in the plane all map the exact same distance AA’ is equal to BB”
- Orientation is the same
- Special points: no special points, all points move
Translated segments
All segments that are translated are parallel to each other
Reflection of Y axis rule
Ryaxis (x,y) = (-x,y)
Reflection over the x axis rule
Rxaxis (x,y) = (x,-y)
Reflection over Y=x rule
Ry=x (x,y) = (y,x)
Rotation of 90 degree rule
RO,90(x,y) = (-y,x)
Rotation of 180 degree rule
(-x,-y)
Rotation of 270 degrees rule
(y,-x)
Composite function
When a sequence of transformations take place
- order matters
- work from closest transformation to the parenthesis out. Work from the inside out
Double reflections of parallel lines
Makes a translation. The translation is double the distance between the parallel lines. Direction of the translation is determined by the order of the reflections
Double reflections over intersecting lines
Makes a rotation. The rotation is double the angle formed between the intersecting lines
Vertical angles
- Are congruent
- Are the two non-adjacent angles formed by intersecting lines
Transversal
A line that crosses parallel lines.
When a transversal crosses parallel lines, alternate exterior angles are congruent and corresponding angles are congruent
- a line that cuts across two or more lines (they do not have to be parallel)
Parallel lines are formed when
We translate an angle along one of its rays. If we extend those rays into lines we form new angles
- we use arrowheads to denote which lines are parallel to each other
Corresponding angles
Corresponding angles are congruent and they are in the same place but on the other parallel line
Alternate exterior angles
They are on alternating sides of the transversal and exterior because they are on the outside of the parallel lines.
-are congruent
Alternate interior angles
Alternate because they are on alternating sides of the transversal and interior because they are on the interior of the parallel lines
-congruent
Consecutive interior angles or same side interior angles
Same side because they are on the same side of the transversal and interior because they are on the interior of the parallel lines
-are supplementary
Consecutive exterior angles or same side exterior angles
Same side because they are on the same side of the transversal and exterior because they are on the exterior of the parallel lines
-supplementary
Congurent terms
Corresponding angles are congruent
Alternate interioir angles are congurent
Alternate exterior angles are congruent
Supplementary terms
Consecutive (same side) interior and exterioir angles are supplementary
Adjacent angles
Angles that share a vertex and a ray and no interior points
- 2 angles that have:
1. same vertex
2. Common side
3. No overlap
Linear pair
Two angles that are adjacent and sum to 180
Supplementary angles
Two angles are supplementary if the sum of their measures is 180. The two angles do not have to be adjacent or touching in any way
-two angles that total 180
Complementary angles
Two angles are complementary if the sum of their measures is 90
- two angles that total 90
What angle do you use
The acute angle and not it’s obtuse supplement
-so that that one could measure it on a regular protractor
Complementary adjacent angles
Make right angles
What if the non-shared sides of adjacent angles are opposite rays
The two angles form a linear pair
Ry=c
Up or down
Rx=c
Left or right
Reflection characteristic
- Distance from pre-image to image
- Orientation
- Special points
- Different- points move different distance depending on how far away they are from the line of reflection
- Different- orientation is reversed
- Points on the line of reflection
What is orientation
The ordering of the points on the shape
What is the only point that doesn’t move with a rotation
The center of rotation
Characteristics of Rotation
- Distance from pre-image to image
- Orientation
- Special points
- Different- move points along an arc and the further the point is from the further it travels
- Same
- Center of rotation- a fixed point
Positive rotations are
Counter clockwise
Negative rotations are
Clockwise
Charcteristics of a translation
- Distance from pre-image to image
- Orientation
- Special points
- Same- due to the parallel relationships found in a translation all points move the exact same distances
- same
- None- All points move a fixed distance and direction
Composite function
When a sequence of transformations take place. Work from the inside out
A double reflection over parallel lines is a
Translation. The translation is double the distance between the parallel lines. Order determines direction
A double reflection over intersecting lines is a
Rotation. The rotation is double the angle formed between intersecting lines. Direction is determined by the order of reflections
Rigid motions
Preserve the shape and size of pre-image and image. AKA isometric transformations
-Isometric transformations are congruent
Congruence of figures is defined by
Two figures are congruent if and only if one can be mapped onto the other by one or more rigid motions
-Dialations and stretches are not congruent
How do you know if two figures are congruent or not
When one figure can be moved to fit exactly on another
Congruence statement
Relates one identical object to another by identify the corresponding parts that match each other
CPCFC
Corresponding parts of congruent figures are congruent
Congruence of triangles is defined by
Two triangles are congruent if and only if one can be mapped onto the other by one or more rigid motions
AS1S2, when S2 is greater than S1
This forms a triangle congruence relashionship. There is only one way for S2 to be placed to complete the triangle
AS1S2, when S2 is less than S1 (too short)
S2 may not be long enough to close the triangle. This is not a congruence relashionship, it doesn’t even form a triangle
AS1S2, when S2 is less than S1 (1 intersection)
Typically known as HL, Hypotenuse leg. It gets this special name because it is the right triangle that locks this shape. This forms a congruence relashionship because in a right triangle, if you know two sides, you can use the pythagorean theorem to calculate the third side. Then you have SSS or SAS. HL forms a triangle congruence
AS1S2, when S2 is less than S1 (2 intersections)
Known as the AMBIGIOUS CASE because two different triangles can be formed by this information. Because S2 is shorter it can swing to form two possible locations. This does NOT form a triangle congruence relashionship
SSS
Side side side. Guarantees congruence in triangles
SAS
Side angle side. Establishes/ guarantees congruence
ASA/SAA
Angle side angle/Side angle angle. Guarantees congruence in triangles
HLs are
- have a 90 degree angle
- SSA or SAS
As1s2 (s2>s1) are also
ASS or SSA
What do 1 line summetric figures have in common
- 2 congruent parts
- A reflection
What do rotation and refkectional shapes have in common
- 2 line of reflection
- order 2
What kinds of orders do point symmetric figures have
-even orders of at least 2
Write the equation of a vertical line
A. Y axis
B. Line passes through pt 1
C. Line goes through pt 2
A. X=0
B. X=1
C. X=2
Stretch, dilation, translation, reflection, rotation coordinates
Stretch- x, (c)y or vice versa. One stays and the other is multiplies
Dilation- 3X, 3Y- both are multiplied by the same amount
Translation: -X+2, Y+5, moves all points the same distance
