KA Geometry Unit 2: Transformation Properties and Proofs
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62 Terms
What is the sum of the interior angles of a triangle?
180 degrees.
What is the formula for perimeter?
Perimeter = Sum of all sides of a shape.
What is the definition of area?
Area is the total space inside a shape.
What is a transversal in geometry?
A transversal is a line that passes through two lines in the same plane at two distinct points. It creates several angles, including corresponding angles, alternate interior angles, and alternate exterior angles.
What are rigid transformations?
Rigid transformations are transformations where angles, side lengths, and overall shape stay the same.
What properties are preserved during a rotation?
Side lengths, perimeter, area, shape congruence, and angle measures are preserved during a rotation.
What properties are preserved during a reflection?
Side lengths, angle measures, perimeter, area, and shape congruence are preserved.
What properties are not preserved during a reflection?
Coordinates (unless the point is on the line of reflection).
What properties are preserved during rigid transformations?
Lengths of segments, the measures of angles, and the areas of shapes are preserved.
What is dilation?
A dilation changes the size of a shape while preserving its shape, but it does not preserve distances between points.
What properties are not preserved during dilation?
Coordinates of vertices, lengths of line segments, perimeter, and area.
What properties are preserved during dilation?
Angle measures, parallelism of line segments, and shape similarity.
What happens to lines not going through the center of dilation?
The distances of every point on that line from the center are scaled, creating a new line parallel to the original.
What happens to lines passing through the center of dilation?
The line remains unchanged, but points on it move closer to or farther from the center.
What is a rigid transformation?
A transformation where side lengths and angle measures remain unchanged.
What is an example of a rigid transformation?
A translation, rotation, or reflection.
What is the key difference between a dilation and a rigid transformation?
A dilation changes the size of a shape, whereas rigid transformations preserve size.
What is preserved during a translation?
Angle measures and segment lengths are preserved during a translation.
What is a vertical stretch?
A vertical stretch changes the size of the figure, so neither angle measures nor segment lengths are preserved.
What are examples of transformations that preserve segment lengths and angle measures?
Translation and reflection preserve both segment lengths and angle measures.
How can transformations be defined in geometric problems?
By describing how points map to other points using specific lines or rotations.
What does “maps to itself” indicate in a geometric problem?
It indicates a reflection where the point lies on the line of reflection.
How do we identify a rotation in a problem?
By looking for clues like a specific center point and a rotation angle (e.g., 137 degrees counterclockwise).
How do we identify a translation in a problem?
By identifying the change in position of a shape, like moving a circle from (x, y) to (x+11, y-7).
What is a counterexample?
A counterexample is an instance that disproves a general statement or rule.
Why are counterexamples used in geometry?
To refine definitions and show where they break down.
How does symmetry relate to transformations?
Symmetry involves transformations where a shape remains unchanged, such as rotations or reflections.
What is reflective symmetry?
Reflective symmetry is when a shape can be divided into two identical halves by a line of symmetry.
What is rotational symmetry?
Rotational symmetry occurs when a shape looks the same after a certain rotation (e.g., 180 degrees).
How do you find the symmetry of a quadrilateral?
Draw the axis of symmetry, reflect the points, connect them, and recognize the shape.
How can we determine the type of transformation from a problem description?
By identifying key phrases, such as “maps to itself” for a reflection or “rotation around point O” for a rotation.
What properties are preserved during rigid transformations?
Lengths of segments, measures of angles, and areas are preserved.
What is the definition of a rotation?
A rotation turns every point in a shape by a certain angle around a fixed point.
What is the transformation notation for a reflection over the y-axis?
rᵧ-axis or rₓ=₀, which flips the x-coordinate.
What is the transformation notation for a rotation of 90° clockwise?
Rₓ(0,0), -90° for a 90° clockwise rotation around the origin.
How are rigid transformations useful in geometric proofs?
They preserve key properties like lengths, angles, and areas, which help prove geometric relationships.
What does similarity in dilations mean?
Similarity means shapes have the same angles but may have different sizes.
How do we prove that corresponding angles are equal using transformations?
By applying rigid transformations and showing that the properties of the shape remain the same.
What is the use of dilations in geometric proofs?
Dilations are used to preserve angle measures and proportional relationships between parts of the shape.
What is a translation’s effect on shapes?
A translation moves every point in the shape by the same distance in the same direction.
What are examples of shapes with symmetry?
Shapes like squares, circles, and isosceles triangles have reflective and/or rotational symmetry.
What is the difference between reflective and rotational symmetry?
Reflective symmetry involves a line of symmetry, while rotational symmetry involves rotating the shape around a point.
How does a vertical stretch affect a shape’s symmetry?
A vertical stretch does not preserve angle measures or segment lengths, disrupting symmetry.
What does a reflection do to a shape?
A reflection flips the shape across a line, preserving size and shape but reversing the coordinates.
What does a rotation do to a shape?
A rotation turns a shape around a fixed point, preserving angles and side lengths.
What is the formula for a dilation with a scale factor?
Each coordinate of the shape is multiplied by the scale factor.
What is an axis of symmetry?
An axis of symmetry is a line that divides a shape into two identical halves.
What is the formula for a reflection over the x-axis?
(x, y) → (x, -y).
What does “maps to a new point” indicate in a problem?
It suggests a transformation, such as a rotation or reflection.
What properties do rotations preserve?
Rotations preserve shape congruence, angle measures, side lengths, and area.
What is the effect of a translation on coordinates?
A translation shifts all points in a shape by a constant amount in the same direction.
How do you use rigid transformations in proofs?
To show that certain properties, like congruence or equality of angles, are preserved during transformations.
What does it mean for a shape to have symmetry?
A shape has symmetry if it can be transformed (reflected or rotated) and still look the same.
How does symmetry relate to transformations?
Symmetry shows that a shape will remain the same after a transformation like rotation or reflection.
What is the effect of a dilation on parallel lines?
Dilation preserves parallelism between lines, even as the shape changes size.
How does dilation affect areas?
The area of a dilated shape is multiplied by the square of the scale factor.
What is preserved in a dilatation?
Angle Measures
Parallelism of Line Segments
Shape Similarity (The shapes have the same angles but different sizes
What is not preserved in a dilatation?
Coordinates of the vertices (coordinates will change as it shrinks/grows)
Length of line segments/side lengths (multiplied by scale factor)
Permeter (multiplied by scale factor)
Area (multiplied by scale factor)
What is preserved in ALL Rigid Transformations?
Side Lengths
Angle measures
Area
Shape Congruence
Perimeter
What is not preserved in Rigid Transformations?
Coordinates (in most cases): Unless the point being reflected lies on the line of reflection, its coordinates will change.
Why are side lengths, angle measures, perimeter, area and congruence preserved in rigid transformations?
Because rigid transformations maintain the original shape and size of the figure, ensuring that distances and angles remain constant.
What does the magnitude of rotation mean?
The magnitude of the rotation is the number of degrees we can rotate the figure to map it onto itself.