KA Geometry Unit 2

Unit 2: Transformation Properties and proofs


Rigid transformations overview: 

All sides of a triangle add to 180 degrees.


Perimeter = Sum of all sides of a shape

Area: Total area of a shape


In geometry, a transversal is a line that passes through two lines in the same plane at two distinct points. 





To find measures using rigid transformations, we need to know what rigid transformations are.

Rigid Transformations are transformations where, when angles transform, the angles measurement, side length, and everything stays the same.


For example (not relating to the paragraph above), if we had a problem asking us to solve the missing length of a triangle, we know that all sides of a triangle add up to 180 degrees, so we solve by finding using the angle lengths we already know, and doing the math for finding the missing one.


What Properties of a Shape are preserved during transformations (here its rotations and reflections).


In a rotation, the things that are preserved are the:


  • Side Lengths (Radius Length in Circle) 

  • Perimeter (Circumference in circle) (we also know since the radius is preserved the circumference will be as well. (dependent) 

  • Area (we also know since radius is preserved (dependent)

  • Shape Congruence (the shape remains congruent to the original. A rotation simply turns the shape; it doesn't change its size or shape. )

  • Angle Measures (the shapes angles stay the same, the rotation simply rotates the shape not changes the measurement of angles.



In a reflection, the properties that are preserved are the:


  • Side Lengths

  • Angle measures: The angles within the shape remain the same.

  • Perimeter: Since side lengths are preserved, the perimeter is also preserved.

  • Area: The area enclosed by the shape remains the same.

  • Shape congruence: The reflected image is congruent (identical in shape and size) to the original shape.


The properties that are not preserved are:


  • Coordinates (in most cases): Unless the point being reflected lies on the line of reflection, its coordinates will change.



All rigid transformations preserve::


1. lengths of segments, 

2. the measures of angles, 

3. the areas of shapes. 


In short, side lengths, angle measurements, perimeter, area, always stay the same no matter the rigid transformation.




Mapping Shapes

Say you were given four sequences that give the instructions of rigid transformations. You were asked which one correctly maps a shape triangle to the hypothetical transformed shape.


To answer this, you simply just do the steps for each sequence. The sequence(s) that maps up the shape correctly is your answer.

Dilation preserved properties: 


In a dilation, that properties that aren’t preserved are


  • The coordinates of the vertices are not preserved in a dilation (since it shrinks/increases (coordinates will change) 

  • Length of line segments (side lengths) because lengths are multiplied by the scale factor, but they will stay parallel.

  • Perimeter: The perimeter is multiplied by the scale factor.

  • Area: The area is multiplied by the square of the scale factor.



In a dilation, properties that are preserved are:


  • Angle measures: Angles remain congruent.

  • Parallelism of line segments: Corresponding line segments remain parallel.

  • Shape (similarity): The dilated figure is similar to the original figure. Similarity means that the shapes have the same angles, but different sizes.



Lines NOT going through the center of dilation: Imagine a line segment not touching the center of dilation.. Dilation expands or contracts the distance of every point on that line from the center. This creates a new line segment with a different length but the same direction—hence, parallel. It will be on a new line.


Lines not going through the center of dilation: If a line segment already passes through the center of dilation, scaling the distances of its points from the center simply extends or shortens the same line. The line remains unchanged (though points on it move). It will be on the same line.




Properties & definitions of transformations: 


In rigid transformations the side lengths and angle measures are always preserved. It doesn’t matter what kind of rigid transformation it is (rotation, reflection, or translation) the side lengths and angle measures WILL ALWAYS stay the same.

A dilation IS NOT a rigid transformation. Adilation changes the size of a figure while preserving its shape, which means it does not maintain the original distances between points, a key characteristic of a rigid transformation.


In a dilation, only the angle measures are preserved, not the side lengths.


What is preserved with a sequence of transformations? Example of a sequence of transformations:


A translation

A reflection over a horizontal line PQ

A vertical stretch about PQ


We can use what we know is preserved and not preserved for each transformation. For this example, lets see whether angle measures and segment lengths are preserved:


  1. A translation - both angle measures and segment lengths are preserved

  2. A reflection over a horizontal line PQ - both angle measures and segment lengths are preserved

  3. A vertical stretch about PQ - neither angle measures or segment lengths are preserved


With a vertical stretch, neither angle measures or side lengths will be preserved, as you are stretching the shape (changes the side lengths and the measures of the angles).


So therefore, the sequence of transformations doesn’t preserve either the angle measures or side lengths.


Therefore, a vertical stretch is not a rigid transformation, just like how a dilation IS NOT A RIGID TRANSFORMATION.


To define transformations, you have to first try to imagine the problem in your head. For example, if the problem is saying each point on a “certain line” maps to itself. 


This already tells us that we are dealing with a reflection, because if a point sits on the line of a reflection, it's not going to move, so it will “map to itself”


Words that say a reflection:


“Each point on line (3x = y + 2) maps to itself”

“Any point that isn’t on (3x = y + 2) maps to a new point P’ that the perpendicular bisector of the (original point and new point) is the line of reflection (y = 3x -2)


Words that say a rotation :


“Point O maps to itself”


“Every point V on a circle Centered at O maps to a new point W on Circle C so that the counterclockwise angle from line segment OV to line segment OW measures 137 degrees”


Here, we have point O. There is a circle around point O. We are rotating something around point O counterclockwise.


It’s easy to indicate what kind of rigid transformation it is after reading the problem since it mentions rotating angles.


Words that say a translation:


“Each circle O with radius r and centered at (x,y) is mapped to a circle O’ with radius r and centered at (x+11, y-7)


This above is clearly a translation. If we read closely, the (x+11, y-7) clearly shows this.


To identify the type of transformation, you just need to know how all the rigid transformations work, and how they would work for two points.


Problem Examples:


Which counterexample shows that Liam's definition does not fully define a translation?


Problems like these give you a transformation then asks for a counterexample that matches Liam’s definition but is not the right way a transformation is supposed to be. 


This challenges us to precisely define transformations we are doing.


Symmetry:


Reflective Symmetry

 

The axis of symmetry is an imaginary line that divides a shape into two identical halves, where each side is a mirror image of the other. 


Basically, you could fold the shape along the line and the two halves would match up.


To determine whether a line is the axis of symmetry, we can try to reflect points in the shape along the line. 


If the points are the same distance, the line is the axis of symmetry, if they don’t then it is not an axis of symmetry.


Rotational Symmetry

To see if two shapes are symmetric, we can imagine one shape rotated by a certain amount of degrees.


The easiest way would be to rotate it by 180 degrees. If the shape looks the same, it is rotationally symmetric. If it isn’t, it is NOT rotationally symmetric.


Essentially, A figure has rotational symmetry if there is a rotation that maps the figure onto itself.


Finding a quadrilateral from its symmetries


To identify a quadrilateral from its symmetries, you first:


  1. Draw axis of symmetry that is given

  2. Draw in the points given

  3. Reflect the points off the axis of symmetry

  4. Connect the points

  5. Recognize the shape from there.


If you were asked to find a quadrilateral based on two perpendicular lines, you should use the points you are given to draw points on both of the lines you were given using the slopes line.

Proofs with transformations:

If you were asked to perform a transformation that proves that corresponding angles are always equal, or something else for example, you would need a proof.

We can use the multiple-choice answers and do the process of elimination until we get the correct answer.


Another example: You were asked which statement proves vertical angles are always equal.

Here, we need to find if the measures of two angles are equal.


Why are transformations useful in writing geometric proofs?

We often use rigid transformations and dilations in geometric proofs because they preserve certain properties. 


Rigid transformations—such as translations, rotations, and reflections—preserve the lengths of segments, the measures of angles, and the areas of shapes. 

2. the measures of angles, 

3. the areas of shapes. 


 Dilations, on the other hand, change the size of a shape, but they preserve the measures of angles, the proportions, and relationships between different parts of the shape.


Certain transformations preserve even more properties. 


For example, when we translate or dilate a figure, figures that were parallel before the transformation are still parallel after it. 


So, when we use rigid transformations and dilations in geometric proofs, we can be confident that certain properties will remain the same even when the shapes are moved or resized.


What is a counterexample?


A counterexample is a specific instance (usually a mathematical example) that disproves a general statement or rule. Counterexamples are often used to refine definitions, to show where they break down, and to help us determine how to make them stronger.


For example, we might want to define a rotation of a geometric figure as a transformation that turns every point in the figure the same number of degrees around a center point. 


But, if we think about it, we can come up with a counterexample that shows this definition isn't quite right. If some points turn the same number of degrees, but in a different direction, we don't get a rotation.

So, using this counterexample, we might revise our definition to say that a rotation is a transformation of a geometric figure that turns every point in the figure the same number of degrees and in the same direction around a center point.


Counterexamples are a powerful tool in mathematics and can help us write better, more accurate definitions.


How does symmetry relate to transformations?


Symmetry is all about how a shape looks the same after being transformed in some way. A shape has rotational symmetry if it looks the same after being rotated around a point. It has reflective symmetry if it looks the same after being reflected across a line.


So, can a figure have one type of symmetry but not the other?


Absolutely! For example, an isosceles trapezoid has reflective symmetry across its vertical line of symmetry, but it doesn't have rotational symmetry. 


On the flip side, a parallelogram that is not a rhombus has rotational symmetry (it looks the same after being rotated 180 degrees, but it doesn't have reflective symmetry.


How can we notate the transformations?


We can define transformations more precisely using mathematical notation.


For a translation, we specify the displacement vector that describes the direction and magnitude of the movement. For example, we could write T (5,2) to represent moving an object 

5 units to the right and 2 units up. We can also write a mapping function for the same translation: (x,y) becomes (x + 5 , y+2)


For a reflection, we specify the line of symmetry. For example, we could write reflecting over the y-axis as rᵧ-axis. (y-axis). We could also use the equation of the line of reflection, such as rₓ=₀. In this case, the y-coordinate stays the same, but the x-coordinate flips to the opposite side of the y-axis.   (x, y) → (-x, y)

For a rotation, we specify the point around which we rotate, the angle, and the direction. For example, we can describe rotating around the origin by 90° clockwise as
Rₓ(0,0), -90°. 

We'll be using positive angles for counterclockwise rotations and negative angles for clockwise rotations. Rotations around the origin in multiples of 90° have straightforward mappings. Here is a mapping function for the rotation. (x, y) → (y, -x)

Rotations by other angles involve trigonometry, so we'll save those for a later course.

The magnitude of the rotation is the number of degrees we can rotate the figure to map it onto itself.