EnVision Geometry 3-1: Reflections

Name: EnVision Geometry 3-1: Reflections
Uploaded: 2021-10-25T06:50:37.000Z
Duration: 1 h 2 min 41 s
Description: Envision Geometry

Transformations in Geometry: Rigid Motions

Overview of Rigid Motions

The topic introduces the four types of rigid motion: reflections, translations, rotations, and their properties.

It emphasizes that transformations can be combined to create new images and proofs, including demonstrating that a composition of two or more rigid motions is also a rigid motion.

Lessons and Vocabulary

There are five lessons in this topic with six new vocabulary terms: composition of rigid motions, glide reflection, point symmetry, and reflectional symmetry.

The lesson aims to help students find reflected images and write rules for reflections while defining reflection as a transformation across a line.

Exploring Reflections

An illustration shows irregular pentagon-shaped tiles; students are asked to identify which tiles are copies of tile one based on color and shape.

Discussion about how some tiles appear as mirror images due to being flipped across a line suggests they are not exact copies but rather reflections.

Essential Questions on Reflection Properties

Students are prompted to consider how the properties of reflection transform figures. A key definition states that a rigid motion preserves length and angle measures.

An example checks if an image preserves length by comparing corresponding sides; if lengths differ, it cannot be classified as a rigid motion.

Rigid Motion Characteristics

Example analysis confirms whether shapes preserve both length and angle measures; if they do, they qualify as rigid motions despite points being in reverse order.

A distinction is made between examples where size changes indicate non-rigid transformations versus those that maintain dimensions.

Defining Reflection

Reflection is defined as transforming each point across a line of reflection. If points lie on the line, they remain unchanged; otherwise, perpendicular bisectors determine their positions post-reflection.

Notation for reflected points uses primes (e.g., triangle ABC becomes triangle A'B'C'), reinforcing that reflections are indeed rigid motions preserving lengths and angles.

Reflecting Figures Across Lines

Reflection of Triangles and Quadrilaterals

Steps to Reflect Shapes Across Lines

To reflect a shape, draw a perpendicular line from the original shape to the line of reflection. Then, create another line that is equidistant from the first line and also perpendicular.

For triangle L and N, follow similar steps as in previous examples to find the reflection across line N. Draw a perpendicular line while maintaining equal distance from the original triangle.

Reflecting Points on a Coordinate Plane

Example involves reflecting quadrilateral FGHJ with coordinates (0,3), (2,4), (4,2), and (-2,0). The task is to determine their reflected points.

When reflecting across the x-axis, maintain the same x-coordinate while changing the y-coordinate to its opposite value. For instance, point (4,2) becomes (4,-2).

Understanding Reflections Across Axes

Key insight: when reflecting points across the x-axis, x-values remain unchanged while y-values become their opposites. For example:

Point (3,-3) reflects to (3,3)

Point (0,y) remains at (0,y)

Reflecting across the y-axis means keeping y-values constant but changing x-values to their opposites. For instance:

Point (-2,0) reflects to (2,0).

Practical Reflection Examples

Task involves reflecting triangle ABC across both axes. Students are encouraged to graph pre-image points before performing reflections.

After plotting points A(5,-6), B(1,-2), C(-3,-4), students should identify how coordinates change during reflections.

General Rules for Reflections

General rule: for any point P(x,y) :

Reflection across x-axis results in P'(x,-y) .

Reflection across y-axis results in P'(-x,y).

To describe reflections mathematically:

Find midpoints between original and reflected points; they lie on the line of reflection.

Use slope formula m = y_2-y_1/x_2-x_1 for determining equations of lines based on two points.

Reflection and Midpoints in Geometry

Understanding Reflection Lines

The concept of reflecting an image across a line, specifically the line y = x, is introduced as a foundational step in geometry.

To find the equation of the line of reflection, one must first calculate the midpoint between two points (e.g., C and C prime).

The midpoint formula involves averaging the x-coordinates and y-coordinates separately to determine the center point between two given coordinates.

Calculating Slopes and Equations

The slope is calculated using the difference in y-coordinates over the difference in x-coordinates; for example, from points derived earlier, it results in a slope of -1.

Using point-slope form, one can derive equations such as y = -x - 5/2, indicating reflections across specific lines like y = -x.

Finding Additional Midpoints

For part B, midpoints are again calculated for different pairs (F and F prime), leading to further understanding of how these calculations relate to reflections.

Careful attention is needed when identifying corresponding points (like G and G prime); errors can lead to incorrect conclusions about slopes.

Vertical Lines and Undefined Slopes

An undefined slope indicates a vertical line; thus, equations take on forms like x = 1, which represents reflections across vertical lines such as x = 1.

Real-world Applications of Reflection

Billiards Example

A practical scenario involving billiards illustrates how angles of incidence equal angles of reflection when aiming at balls on a table.

The cue ball's trajectory must be carefully planned so that it reflects off surfaces without hitting obstacles (like yellow balls).

Drawing Reflections

To visualize this problem effectively, drawing segments from both cue ball positions helps clarify where reflections occur relative to lines.

Congruent Angles

Observing that angles formed by the path of the cue ball and its reflection are congruent reinforces geometric principles regarding symmetry.

Student Reflection Across Mirrors

Diagramming Reflections

In another example involving students observing their images in mirrors, creating accurate diagrams helps illustrate actual versus perceived positions.

Reflection Transformations in Geometry

Understanding Reflections

A reflection is defined as a type of transformation that mirrors each point in the pre-image across a designated line of reflection.

Points located directly on the line of reflection remain unchanged, reflecting onto themselves without movement.

For points not situated on the line of reflection, one can draw a perpendicular line to find their corresponding image by maintaining equal distance on the opposite side.