geometry reflections homework

Reflections

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New York State Common Core Math Geometry, Module 1, Lesson 14

Worksheets for Geometry

Student Outcomes

• Students learn the precise definition of a reflection.
• Students construct the line of reflection of a figure and its reflected image.
• Students construct the image of a figure when provided the line of reflection.

Exploratory Challenge

Think back to Lesson 12 where you were asked to describe to your partner how to reflect a figure across a line. The greatest challenge in providing the description was using the precise vocabulary necessary for accurate results. Let’s explore the language that yields the results we are looking for.

△ 𝐴𝐵𝐶 is reflected across 𝐷𝐸 and maps onto △ 𝐴′𝐵′𝐶′.

Use your compass and straightedge to construct the perpendicular bisector of each of the segments connecting 𝐴 to 𝐴′, 𝐵 to 𝐵′, and 𝐶 to 𝐶′. What do you notice about these perpendicular bisectors?

Label the point at which 𝐴𝐴′ intersects 𝐷𝐸 as point 𝑂. What is true about 𝐴𝑂 and 𝐴′𝑂? How do you know this is true?

You just demonstrated that the line of reflection between a figure and its reflected image is also the perpendicular bisector of the segments connecting corresponding points on the figures. In the Exploratory Challenge, you were given the pre-image, the image, and the line of reflection. For your next challenge, try finding the line of reflection provided a pre-image and image.

Construct the segment that represents the line of reflection for quadrilateral 𝐴𝐵𝐶𝐷 and its image 𝐴′𝐵′𝐶′𝐷′.

What is true about each point on 𝐴𝐵𝐶𝐷 and its corresponding point on 𝐴′𝐵′𝐶′𝐷′ with respect to the line of reflection?

Notice one very important fact about reflections. Every point in the original figure is carried to a corresponding point on the image by the same rule—a reflection across a specific line. This brings us to a critical definition:

REFLECTION : For a line 𝑙 in the plane, a reflection across 𝑙 is the transformation 𝑟𝑙of the plane defined as follows:

• For any point 𝑃 on the line 𝑙, 𝑟𝑙 (𝑃) = 𝑃, and
• For any point 𝑃 not on 𝑙, 𝑟𝑙 (𝑃) is the point 𝑄 so that 𝑙 is the perpendicular bisector of the segment 𝑃𝑄.

If the line is specified using two points, as in 𝐴𝐵 , then the reflection is often denoted by 𝑟𝐴𝐵. Just as we did in the last lesson, let’s examine this definition more closely:

• A transformation of the plane—the entire plane is transformed; what was once on one side of the line of reflection is now on the opposite side;
• 𝑟𝑙(𝑃) = 𝑃 means that the points on line 𝑙 are left fixed—the only part of the entire plane that is left fixed is the line of reflection itself;
• 𝑟𝑙(𝑃) is the point 𝑄—the transformation 𝑟𝑙 maps the point 𝑃 to the point 𝑄;
• The line of reflection 𝑙 is the perpendicular bisector of the segment 𝑃𝑄—to find 𝑄, first construct the perpendicular line 𝑚 to the line 𝑙 that passes through the point 𝑃. Label the intersection of 𝑙 and 𝑚 as 𝑁. Then locate the point 𝑄 on 𝑚 on the other side of 𝑙 such that 𝑃𝑁 = 𝑁𝑄.

Examples 2–3 Construct the line of reflection across which each image below was reflected

• You have shown that a line of reflection is the perpendicular bisector of segments connecting corresponding points on a figure and its reflected image. You have also constructed a line of reflection between a figure and its reflected image. Now we need to explore methods for constructing the reflected image itself. The first few steps are provided for you in this next stage.

Example 4 The task at hand is to construct the reflection of △ 𝐴𝐵𝐶 over ̅𝐷𝐸̅̅̅. Follow the steps below to get started; then complete the construction on your own.

• Construct circle 𝐴: center𝐴, with radius such that the circle crosses ̅𝐷𝐸̅̅̅ at two points (labeled 𝐹 and 𝐺).
• Construct circle 𝐹: center 𝐹, radius 𝐹𝐴 and circle 𝐺: center 𝐺, radius 𝐺𝐴. Label the (unlabeled) point of intersection between circles 𝐹 and 𝐺 as point 𝐴′. This is the reflection of vertex 𝐴 across ̅𝐷𝐸̅̅̅.
• Repeat steps 1 and 2 for vertices 𝐵 and 𝐶 to locate 𝐵′ and 𝐶′.
• Connect 𝐴′, 𝐵′, and 𝐶′ to construct the reflected triangle. Things to consider: When you found the line of reflection earlier, you did this by constructing perpendicular bisectors of segments joining two corresponding vertices. How does the reflection you constructed above relate to your earlier efforts at finding the line of reflection itself? Why did the construction above work?

Example 5 Now try a slightly more complex figure. Reflect 𝐴𝐵𝐶𝐷 across 𝐸𝐹̅̅̅̅.

Lesson Summary

• A reflection carries segments onto segments of equal length.
• A reflection carries angles onto angles of equal measure.

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Reflection Worksheets

Our printable reflection worksheets have exclusive pages to understand the concepts of reflection and symmetry. Exercises to graph the images of figures across the line of reflection, reflection of points and shapes are here for practice. In addition, skills to write the coordinates of the reflected images and more are in these pdf worksheets, making them ideal for students of grade 5 through high school. Flip through our free reflection worksheets and catch a glimpse of what’s in store!

Printing Help - Please do not print reflection worksheets directly from the browser. Kindly download them and print.

Reflection across the Axes

In these reflection worksheets the figure and a line of reflection are given in each problem. Draw the image obtained after reflection.

• Download the set

Draw the other Half: Mirror Image

Draw the other half of the symmetrical shape. Students of 5th grade need to practice these pages to know the relationship between reflection and symmetry.

Reflection of a Point

In these printable worksheets for grade 6 and grade 7 reflect the given point and graph the image across the axes and across x=a, y=b, where a and b are parameters.

Choose the Correct Reflection

This practice set tasks 6th grade and 7th grade students to identify the reflection of the given point from the given options. Reflect the point across the line of reflection.

Reflection of Shapes

In these reflection worksheet pdfs, reflect the shapes across the lines of reflection. Each worksheet has eight problems.

Reflection of Triangles

Reflect each triangle and draw its image on the grid following the given rule (across the axes; x=a; y=b) shown above each grid.

Reflection of Quadrilaterals

Reflect each quadrilateral across the given line of reflection. Graph the image on the grid and label them.

Write the Rules

In these printable 8th grade worksheets write a rule to describe each reflection by determining if the reflection across the x-axis, across the y-axis or across a specific line.

Writing Coordinates: With Graph

Graph the image of each figure after the given reflection. Label the image and write the coordinates.

Writing Coordinates

Observe the given coordinates carefully. Write the new coordinates obtained after reflection in these pdf worksheets for grade 8 and high school.

Related Worksheets

» Symmetry

» Slide, Flip and Turn

» Rotation

» Translation

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Reflections are everywhere ... in mirrors, glass, and here in a lake. ... what do you notice ?

Every point is the same distance from the central line !

... and ...

The reflection has the same size as the original image.

The central line is called the Mirror Line :

Can A Mirror Line Be Vertical?

Yes. Here my dog "Flame" shows a Vertical Mirror Line (with a bit of photo editing).

In fact Mirror Lines can be in any direction . Imagine turning the top image in different directions:

A reflection is a flip over a line

You can try reflecting some shapes about different mirror lines here:

"how do i do it myself".

Just approach it step-by-step. For each corner of the shape:

It is common to label each corner with letters, and to use a little dash (called a Prime ) to mark each corner of the reflected image.

Here the original is ABC and the reflected image is A'B'C'

Some Tricks

When the mirror line is the y-axis we change each (x,y) into (−x,y)

Fold the Paper

And when all else fails, just fold the sheet of paper along the mirror line and then hold it up to the light !

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Rotations, Reflections, and Translations Worksheets

When an object moves across a coordinate plane, most likely a geometric shape, we call this change a transformation. These are the three most common transformations that you will find occurring. They all describe a different type of trajectory of movement and specify exactly how an object changes its direction, distance from the origin, and orientation. While most students see the identification of transformations as an exercise it truly is a math, specifically geometry, vocabulary building activity. When you are able to name these changes you are able to speak a new language that others can understand instantly. This series of worksheets and lessons will help you begin to immerse yourself in this new language.

Aligned Standard: High School Geometry - HSG-CO.A.4

• After A Translation Step-by-step Lesson - We rotate a figure and ask you what it would like after a simple counterclockwise rotation.
• Guided Lesson - Rotations, reflections, and clock directions, Oh my!
• Guided Lesson Explanation - I break rotations into 4 simple steps.
• Practice Worksheet - Shapes take up a lot of space that is why you'll see this one spread over 5 pages.
• Matching Worksheet - Match the figure diagrams to the translations.
• Identify Line Reflection 5 Pack - Are these reflections?
• Compositions and Glide Reflections Worksheet Five Pack - Start by identifying reflections and then move on to more involved problems.
• Answer Keys - These are for all the unlocked materials above.

Homework Sheets

These problems always remind me of origami. It looks like a set of instructions.

• Homework 1 - Draw the line segment between the vertex and the point of rotation.
• Homework 2 - Which diagram shows abcde's reflection?
• Homework 3 - Which diagram shows ABCD rotated 20° counter clockwise about A?

Practice Worksheets

These problems require a higher level of thinking on the part of the kids.

• Practice 1 - Use a protractor to draw the angle of rotation.
• Practice 2 - A reflection flips the figure over a line to create a mirror image.
• Practice 3 - Which diagram shows a mirror reflection down?

Math Skill Quizzes

I did get a little carried away with the mirrored reflections. I will add a better spread of problems in a few weeks.

• Quiz 1 - Label each as a reflection, glide, or transformation.
• Quiz 2 - Is this a reflection?
• Quiz 3 - Use a compass to mark the rotated vertex point on the other side of the angle.

What Are Rotations, Reflections, and Translations of Geometric Shapes?

When a geometric shape is shifted or moved, we call that a transformation. Transformation is an important concept in geometry because it helps us describe this and communicate these movements to others. They are comprised of three sub-categories including translations, rotations, and reflections. To understand what these are, you can stand in front of the mirror and observe yourself when you move sideways and turn towards your side. What you observe in the mirror is exactly what rotations, translations, and reflections are in geometry.

Reflection - The most straightforward concept here is that of reflection. It is a common term that we come across very frequently in our everyday lives. So, what is a reflection in geometry? It is flipping of a point or an entire shape over a mirror line. The mirror line serves as a mirror on the graph. A shape and its reflected image are congruent, and both these are equidistant from the mirror line.

Rotation - Rotation is another term that we use almost every day in our lives. When tightening or loosening a screw you are rotating the screw at an axis of rotation. Similarly, in geometry, rotation is when we turn a shape around a fixed point, which serves as the axis of rotation. To rotate a shape on the graph, you will need the angle of rotation and the point of rotation.

Translation - Translation is simple. A shape and its translated image will have the same orientation and only the vertices will move sideways or upwards or downwards relative to the vertices of a shape. Both the original shape and its translated form are congruent. Each vertex covers an equal distance. This selection of worksheets and lessons teach students to identify and process these three common geometric transformations.

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What Is Reflection in Math? Definition, Examples & How-to

Whether you are just starting to explore reflections in math or need to brush up on your geometry, this simple, middle-school-friendly guide is for you.

Read on to find easy-to-follow definitions and explanations, solved examples, and resources to help you learn and master reflections.

What Is Reflection?

Reflection in mathematics is a geometric transformation where a shape or object is flipped across a line, known as the line of reflection . This results in a mirror image that is the same size and shape as the original but appears flipped or mirrored.

You encounter the concept of reflection every time you look in a mirror.

Try it now:

Step in front of a mirror and raise your right hand.

What hand is your mirror image raising?

If you raise your right hand in front of a mirror, your reflection, i.e. your mirror image, raises its left hand.

Transformations in Geometry

Reflection is one of the 4 types of transformations in geometry .

Other types of geometrical transformations are:

• Translation : Moving a shape without rotating or flipping it. It's like sliding the shape in a particular direction.
• Rotation : Turning a shape around a fixed point. It’s like turning a key in a lock.
• Dilation : Resizing a shape by enlarging or shrinking it evenly. It's like stretching or squeezing the shape while keeping its proportions intact, like inflating or deflating a balloon.

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Properties of Reflection

 To truly understand what reflection is and to distinguish it from other geometrical transformations, we need to know its properties.

Reflection has 5 key properties:

• Shape:  The shape of the original figure and its mirror image are identical.
• Size: The size of the original figure and its mirror image are the same.
• Orientation:  The orientation of the original figure and its mirror image are opposite.
• Distance:  The distance between any point on the original figure and the line of reflection is the same as the distance between the corresponding point on the mirror image and the line of reflection.
• Angle: Angles between intersecting lines are the same in both the original figure and its mirror image.

If any of these five properties are missing, the geometrical transformation is either not a reflection at all or is a combination  of reflection and other transformations.

Important Terms in Reflection

Let’s go over some language we use to talk about reflections.

We use these terms to explain how reflection works:

• Line of Reflection:  The imaginary line across which a shape is reflected to create its mirror image. It is also known as the axis of reflection or mirror line.
• Mirror Image: The image formed when you reflect a shape across the line of reflection. It is identical to the original shape but appears reversed.
• Congruent Figures: Two figures that have the same size and shape. In reflection, the original shape and its mirror image are always congruent.
• Symmetry: The property of a shape or object that remains unchanged when reflected across a line, known as a line of symmetry.

How to Do Reflections on the Coordinate Plane

To reflect a point or figure on the coordinate plane, we sometimes use the X-axis or Y-axis as the line of reflection.

Let’s see how each type of reflection works.

How to Do   Reflection Over the X-axis

Reflection over the x-axis is a transformation where each point in a shape or a graph is flipped across the x-axis .

If you have a point (x, y), reflecting it over the x-axis will give you the point (x, -y).

In other words, the x-coordinate (how far left or right the point is) stays the same, but the y-coordinate (how far up or down the point is) becomes its opposite.

For example, if you have the point (2, 3), reflecting it over the x-axis would give you (2, -3), because the x-coordinate remains 2, but the y-coordinate changes from 3 to -3, flipping it across the x-axis.

How to Do Reflection Over the Y-axis

In reflection over the Y-axis, each point in a shape or graph is mirrored horizontally along the Y-axis .

When you have a point (x, y), reflecting it over the Y-axis gives you the point (-x, y). To put it simply, the y-coordinate remains unchanged, but the x-coordinate changes to its negative value.

For example, let's take the point (4, -5). Reflecting it over the Y-axis results in (-4, -5).

Notice that while the y-coordinate remains -5, the x-coordinate changes from 4 to -4 as it mirrors along the Y-axis.

Here’s a visual representation of reflection over the Y-axis.

How to Do Reflection Over the Line y = x

The line y = x represents all the points where the y-coordinate is equal to the x-coordinate.

It's a diagonal line that passes from the bottom left corner to the top right corner through the origin at a 45-degree angle.

Reflection over the y = x line means flipping a point or shape across this diagonal line .

During this reflection, the x-coordinate of each point becomes its y-coordinate, and the y-coordinate becomes its x-coordinate.

For example, if you have a point (x, y), its reflection over the y = x line would be (y, x).

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Geometry Worksheets

Welcome to the geometry worksheets page at Math-Drills.com where we believe that there is nothing wrong with being square! This page includes Geometry Worksheets on angles, coordinate geometry, triangles, quadrilaterals, transformations and three-dimensional geometry worksheets.

Get out those rulers, protractors and compasses because we've got some great worksheets for geometry! The quadrilaterals are meant to be cut out, measured, folded, compared, and even written upon. They can be quite useful in teaching all sorts of concepts related to quadrilaterals. Just below them, you'll find worksheets meant for angle geometry. Also see the measurement page for more angle worksheets. The bulk of this page is devoted to transformations. Transformational geometry is one of those topics that can be really interesting for students and we've got enough worksheets for that geometry topic to keep your students busy for hours.

Don't miss the challenging, but interesting world of connecting cubes at the bottom of this page. You might encounter a few future artists when you use these worksheets with students.

Most Popular Geometry Worksheets this Week

Lines and Angles

In this section, there are worksheets for two of the basic concepts of geometry: lines and angles.

Lines (or straight lines to be precise) in geometry are continuous and extend in both directions to infinity. They have no width, depth or curvature. In math activities, they are often represented by a drawn straight path with some width. To show that they are lines, arrows are drawn on each end to show they extend to infinity. A line segment is a finite section of a line. Line segments are often represented with points at each end of a drawn straight path. Rays start at a point and extend in a straight line to infinity. This is shown with a point at one end of a drawn straight path and an arrow at the other end.

• Identifying Lines, Line Segments and Rays Identify Lines, Segments and Rays

Angles can be classified into six different types. Acute angles are greater than 0 degrees but less than 90 degrees. Right angles are exactly 90 degrees. Obtuse angles are greater than 90 degrees but less than 180 degrees. Straight angles are exactly 180 degrees. Reflex angles are greater than 180 degrees but less than 360 degrees. Complete/Full angles are exactly 360 degrees.

• Identifying Angle Types Worksheets Identifying Acute and Obtuse Angles Identifying Acute and Obtuse Angles (No Angle Marks) Identifying Acute, Obtuse and Right Angles Identifying Acute, Obtuse and Right Angles (No Angle Marks) Identifying Acute, Obtuse, Right and Straight Angles Identifying Acute, Obtuse, Right and Straight Angles (No Angle Marks) Identifying Acute, Obtuse, Right, Straight and Reflex Angles Identifying Acute, Obtuse, Right, Straight, Reflex and Complete/Full Angles

There are several angle relationships of which students should be aware. Complementary angles are two angles that together form a 90 degree angle; supplementary angles are two angles that together form a 180 degree angle; and explementary angles are two angles that together form a 360 degree angle. Vertical angles are found at line intersections; angles opposite each other are equal. Students can practice determining and/or calculating the unknown angle(s) in the following angle relationships worksheets.

• Angle Relationships Worksheets Complementary Angles Complementary Angles (Diagrams Rotated) Supplementary Angles Supplementary Angles (Diagrams Rotated) Mixed Complementary and Supplementary Angles Questions (Diagrams Rotated) Explementary Angles Explementary Angles (Diagrams Rotated) Mixed Adjacent Angles Questions (Diagrams Rotated) Vertical/Opposite Angles Vertical/Opposite Angles (Diagrams Rotated) Mixed Angle Relationships Questions(Diagrams Rotated)
• Angles of Transversals Intersecting Parallel Lines Interior Alternate Angles Exterior Alternate Angles Alternate Angles Corresponding Angles Co-Interior Angles Transversals

Measuring angles worksheets, can be found on the Measurement Page

Triangles, Quadrilaterals and Other Shapes

The quadrilaterals set can be used for a number of activities that involve classifying and recognizing quadrilaterals or for finding the properties of quadrilaterals (e.g. that the interior angles add up to 360 degrees). The tangram printables are useful in tangram activities. There are several options available for the tangram printables depending on your printer, and each option includes a large version and smaller versions. If you know someone with a suitable saw, you can use the tangram printable as a template on material such as quarter inch plywood; then simply sand and paint the pieces.

• Shape Sets Quadrilaterals Set Tangrams
• Identifying Regular Polygons Identifying Regular Shapes from Triangles to Octagons

Worksheets for classifying triangles by side and angle properties and for working with Pythagorean theorem.

If you are interested in students measuring angles and sides for themselves, it is best to use the versions with no marks. The marked versions will indicate the right and obtuse angles and the equal sides.

• Classifying Triangles Worksheets Classifying Triangles by Side Properties Classifying Triangles by Angle Properties Classifying Triangles by Side and Angle Properties Classifying Triangles by Side Properties (No Marks) Classifying Triangles by Angle Properties (No Marks) Classifying Triangles by Side and Angle Properties (No Marks)

A cathetus (plural catheti) refers to a side of a right-angle triangle other than the hypotenuse.

• Calculating Triangle Dimensions Using Pythagorean Theorem Calculate the Hypotenuse Using Pythagorean Theorem (No Rotation) Calculate the Hypotenuse Using Pythagorean Theorem Calculate a Cathetus Using Pythagorean Theorem (No Rotation) Calculate a Cathetus Using Pythagorean Theorem Calculate any Side Using Pythagorean Theorem (No Rotation) Calculate any Side Using Pythagorean Theorem

Trigonometric ratios are useful in determining the dimensions of right-angled triangles. The three basic ratios are summarized by the acronym SOHCAHTOA. The SOH part refers to the ratio: sin(α) = O/H where α is an angle measurement; O refers the length of the side (O)pposite the angle measurement and H refers to the length of the (H)ypotenuse of the right-angled triangle. The CAH part refers to the ratio: cos(α) = A/H where A refers to the length of the (A)djacent side to the angle. The TOA refers to the ratio: tan(α) = O/A.

• Calculating Angles and Sides Using Trigonometric Ratios Calculating Angles Using the Sine Ratio Calculating Sides Using the Sine Ratio Calculating Angles and Sides Using the Sine Ratio Calculating Angles Using the Cosine Ratio Calculating Sides Using the Cosine Ratio Calculating Angles and Sides Using the Cosine Ratio Calculating Angles Using the Tangent Ratio Calculating Sides Using the Tangent Ratio Calculating Angles and Sides Using the Tangent Ratio Calculating Angles Using Trigonometric Ratios Calculating Sides Using Trigonometric Ratios Calculating Angles and Sides Using Trigonometric Ratios

Quadrilaterals are interesting shapes to classify. Their classification relies on a few attributes and most quadrilaterals can be classified as more than one shape. A square, for example, is also a parallelogram, rhombus, rectangle and kite. A quick summary of all quadrilaterals is as follows: quadrilaterals have four sides. A square has 90 degree corners and equal length sides. A rectangle has 90 degree corners, but the side lengths don't have to be equal. A rhombus has equal length sides, but the angles don't have to be 90 degrees. A parallelogram has both pairs of opposite sides equal and parallel and both pairs of opposite angles are equal. A trapezoid only needs to have one pair of opposite sides parallel. A kite has two pairs of equal length sides where each pair is joined/adjacent rather than opposite to one other. A bowtie is sometimes included which is a complex quadrilateral with two sides that crossover one another, but they are readily recognizable. Any other four-sided polygon can safely be called a quadrilateral if it doesn't meet any of the criteria for a more specific classification.

• Classifying Quadrilaterals Classifying Simple Quadrilaterals Classifying All Quadrilaterals Classifying All Quadrilaterals (+ Rotation)

Coordinate Plane Worksheets

Coordinate point geometry worksheets to help students learn about the Cartesian plane.

• Plotting Random Coordinate Points Plotting Coordinate Points in All Quadrants Plotting Coordinate Points in Positive x Quadrants Plotting Coordinate Points in Positive y Quadrants

There are many other Cartesian Art plots scattered around the Math-Drills website as many of them are associated with a holiday. To find them quickly, use the search box.

• Cartesian Art Cartesian Art Maple Leaf
• Coordinate Plane Distance and Area Calculating Pythagorean Distances of Coordinate Points Calculating Perimeter and Area of Triangles on Coordinate Planes Calculating Perimeter and Area of Quadrilaterals on Coordinate Planes Calculating Perimeter and Area of Triangles and Quadrilaterals on Coordinate Planes

Transformations Worksheets

Transformations worksheets for translations, reflections, rotations and dilations practice.

Here are two quick and easy ways to check students' answers on the transformational geometry worksheets below. First, you can line up the student's page and the answer page and hold it up to the light. Moving/sliding the pages slightly will show you if the student's answers are correct. Keep the student's page on top and mark it or give feedback as necessary. The second way is to photocopy the answer page onto an overhead transparency. Overlay the transparency on the student's page and flip it up as necessary to mark or give feedback.

Also known as sliding, translations are a way to mathematically describe how something moves on a Cartesian plane. In translations, every vertex and line segment moves the same, so the resulting shape is congruent to the original.

• Translations Worksheets Translation of 3 vertices by up to 3 units. Translation of 3 vertices by up to 6 units. Translation of 3 vertices by up to 25 units. Translation of 4 vertices by up to 6 units. Translation of 5 vertices by up to 6 units.
• Translations Worksheets (Multi-Step) Two-Step Translation of 3 vertices by up to 6 units. Two-Step Translation of 4 vertices by up to 6 units. Three-Step Translation of 3 vertices by up to 6 units. Three-Step Translation of 4 vertices by up to 6 units.

Reflect on this: reflecting shapes over horizontal or vertical lines is actually quite straight-forward, especially if there is a grid involved. Start at one of the original points/vertices and measure the distance to the reflecting line. Note that you should measure perpendicularly or 90 degrees toward the line which is why it is easier with vertical or horizontal reflecting lines than with diagonal lines. Measure out 90 degrees on the other side of the reflecting line, the same distance of course, and make a point to represent the reflected vertex. Once you've done this for all of the vertices, you simply draw in the line segments and your reflected shape will be finished.

Reflecting can also be as simple as paper-folding. Fold the paper on the reflecting line and hold the paper up to the light. On a window is best because you will also have a surface on which to write. Only mark the vertices, don't try to draw the entire shape. Unfold the paper and use a pencil and ruler to draw the line segments between the vertices.

• Reflections Worksheets Reflection of 3 Vertices Over x = 0 and y = 0 Reflection of 4 Vertices Over x = 0 and y = 0 Reflection of 5 Vertices Over x = 0 and y = 0 Reflection of 3 Vertices Over Various Lines Reflection of 4 Vertices Over Various Lines Reflection of 5 Vertices Over Various Lines
• Reflections Worksheets (Multi-Step) Two-Step Reflection of 3 Vertices Over Various Lines Two-Step Reflection of 4 Vertices Over Various Lines Three-Step Reflection of 3 Vertices Over Various Lines Three-Step Reflection of 4 Vertices Over Various Lines

Here's an idea on how to complete rotations without measuring. It works best on a grid and with 90 or 180 degree rotations. You will need a blank overhead projector sheet or other suitable clear plastic sheet and a pen that will work on the page. Non-permanent pens are best because the plastic sheet can be washed and reused. Place the sheet over top of the coordinate axes with the figure to be rotated. With the pen, make a small cross to show the x and y axes being as precise as possible. Also mark the vertices of the shape to be rotated. Using the plastic sheet, perform the rotation, lining up the cross again with the axes. Choose one vertex and mark it on the paper by holding the plastic sheet in place, but flipping it up enough to get a mark on the paper. Do this for the other vertices, then remove the plastic sheet and join the vertices with line segments using a ruler.

• Rotations Worksheets Rotation of 3 Vertices around the Origin Starting in Quadrant I Rotation of 4 Vertices around the Origin Starting in Quadrant I Rotation of 5 Vertices around the Origin Starting in Quadrant I Rotation of 3 Vertices around the Origin Rotation of 4 Vertices around the Origin Rotation of 5 Vertices around the Origin Rotation of 3 Vertices around Any Point Rotation of 4 Vertices around Any Point Rotation of 5 Vertices around Any Point
• Rotations Worksheets (Multi-Step) Two-Step Rotations of 3 Vertices around Any Point Two-Step Rotations of 4 Vertices around Any Point Two-Step Rotations of 5 Vertices around Any Point Three-Step Rotations of 3 Vertices around Any Point Three-Step Rotations of 4 Vertices around Any Point Three-Step Rotations of 5 Vertices around Any Point
• Dilations Worksheets Dilations Using Center (0, 0) Dilations Using Various Centers
• Determining Scale Factors Worksheets Determine Scale Factors of Rectangles (Whole Numbers) Determine Scale Factors of Rectangles (0.5 Intervals) Determine Scale Factors of Rectangles (0.1 Intervals) Determine Scale Factors of Triangles (Whole Numbers) Determine Scale Factors of Triangles (0.5 Intervals) Determine Scale Factors of Triangles (0.1 Intervals) Determine Scale Factors of Rectangles and Triangles (Whole Numbers) Determine Scale Factors of Rectangles and Triangles (0.5 Intervals) Determine Scale Factors of Rectangles Triangles (0.1 Intervals)
• Mixed Transformations Worksheets (Multi-Step) Two-Step Transformations Three-Step Transformations

Constructions Worksheets

Constructions worksheets for constructing bisectors, perpendicular lines and triangle centers.

It is amazing what one can accomplish with a compass, a straight-edge and a pencil. In this section, students will do math like Euclid did over 2000 years ago. Not only will this be a lesson in history, but students will gain valuable skills that they can use in later math studies.

• Constructing Midpoints And Bisectors On Line Segments And Angles Midpoints on Horizontal Line Segments Perpendicular Bisectors on Horizontal Line Segments Perpendicular Bisectors on Rotated Line Segments Angle Bisectors (Angles not Rotated) Angle Bisectors (Angles Randomly Rotated)
• Constructing Perpendicular Lines Construct Perpendicular Lines Through Points on a Line Segment Construct Perpendicular Lines Through Points Not on Line Segment Construct Perpendicular Lines Through Points on Line Segment (Segments are randomly rotated) Construct Perpendicular Lines Through Points Not on Line Segment (Segments are randomly rotated)
• Constructing Triangle Centers Centroids for Acute Triangles Centroids for Mixed Acute and Obtuse Triangles Orthocenters for Acute Triangles Orthocenters for Mixed Acute and Obtuse Triangles Incenters for Acute Triangles Incenters for Mixed Acute and Obtuse Triangles Circumcenters for Acute Triangles Circumcenters for Mixed Acute and Obtuse Triangles All Centers for Acute Triangles All Centers for Mixed Acute and Obtuse Triangles

Three-Dimensional Geometry

Three-dimensional geometry worksheets that are based on connecting cubes and worksheets for classifying three-dimensional figures.

Connecting cubes can be a powerful tool for developing spatial sense in students. The first two worksheets below are difficult to do even for adults, but with a little practice, students will be creating structures much more complex than the ones below. Use isometric grid paper and square graph paper or dot paper to help students create three-dimensional sketches of connecting cubes and side views of structures.

• Connecting Cube Structures Side Views of Connecting Cube Structures Build Connecting Cube Structures
• Classifying Three-Dimensional Figures Classify Prisms Classify Pyramids Classify Prisms and Pyramids

This section includes a number of nets that students can use to build the associated 3D solids. All of the Platonic solids and many of the Archimedean solids are included. A pair of scissors, a little tape and some dexterity are all that are needed. For something a little more substantial, copy or print the nets onto cardstock first. You may also want to check your print settings to make sure you print in "actual size" rather than fitting to the page, so there is no distortion.

• Nets of Three-Dimensional Figures Nets of Platonic and Archimedean Solids Nets of All Platonic Solids Nets of Some Archimedean Solids Net of a Tetrahedron Net of a Cube Net of an Octahedron Net of a Dodecahedron (Version 1) Net of a Dodecahedron (Version 2) Net of an Icosahedron Net of a Truncated Tetrahedron Net of a Cuboctahedron Net of a Truncated Cube Net of a Truncated Octahedron Net of a Rhombicuboctahedron Net of a Truncated Cuboctahedron Net of a Snub Cube Net of an Icosidodecahedron

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Unit 10: Transformations

About this unit.

In this topic you will learn about the most useful math concept for creating video game graphics: geometric transformations, specifically translations, rotations, reflections, and dilations.

You will learn how to perform the transformations, and how to map one figure into another using these transformations.

Introduction to rigid transformations

• Rigid transformations intro (Opens a modal)
• Translations intro (Opens a modal)
• Rotations intro (Opens a modal)
• Identify transformations 4 questions Practice

Translations

• Translating shapes (Opens a modal)
• Determining translations (Opens a modal)
• Translation challenge problem (Opens a modal)
• Properties of translations (Opens a modal)
• Translations review (Opens a modal)
• Translate points 4 questions Practice
• Determine translations 4 questions Practice
• Translate shapes 4 questions Practice
• Rotating shapes (Opens a modal)
• Determining rotations (Opens a modal)
• Rotating shapes about the origin by multiples of 90° (Opens a modal)
• Rotations review (Opens a modal)
• Rotating shapes: center ≠ (0,0) (Opens a modal)
• Rotate points 4 questions Practice
• Determine rotations 4 questions Practice
• Rotate shapes 4 questions Practice
• Rotate shapes: center ≠ (0,0) 4 questions Practice

Reflections

• Reflecting shapes: diagonal line of reflection (Opens a modal)
• Determining reflections (advanced) (Opens a modal)
• Reflecting shapes (Opens a modal)
• Reflections review (Opens a modal)
• Reflect points 4 questions Practice
• Determine reflections 4 questions Practice
• Determine reflections (advanced) 4 questions Practice
• Reflect shapes 4 questions Practice
• Advanced reflections 4 questions Practice

Rigid transformations overview

• No videos or articles available in this lesson
• Find measures using rigid transformations 4 questions Practice
• Rigid transformations: preserved properties 4 questions Practice
• Mapping shapes 4 questions Practice
• Performing dilations (Opens a modal)
• Dilating shapes: shrinking by 1/2 (Opens a modal)
• Dilating shapes: expanding (Opens a modal)
• Dilate points 4 questions Practice
• Dilations: scale factor 4 questions Practice
• Dilations: center 4 questions Practice
• Dilate triangles 4 questions Practice
• Dilations and properties 4 questions Practice

Properties and definitions of transformations

• Precisely defining rotations (Opens a modal)
• Identifying type of transformation (Opens a modal)
• Sequences of transformations 4 questions Practice
• Defining transformations 4 questions Practice
• Intro to reflective symmetry (Opens a modal)
• Intro to rotational symmetry (Opens a modal)
• Finding a quadrilateral from its symmetries (Opens a modal)
• Finding a quadrilateral from its symmetries (example 2) (Opens a modal)
• Reflective symmetry of 2D shapes 4 questions Practice

Old transformations videos

• Performing translations (old) (Opens a modal)
• Performing rotations (old) (Opens a modal)
• Performing reflections: rectangle (old) (Opens a modal)
• Performing reflections: line (old) (Opens a modal)
• Determining translations (old) (Opens a modal)
• Rotation examples (old) (Opens a modal)
• Determining rotations (old) (Opens a modal)
• Dilating lines (Opens a modal)