Since rigid transformations preserve distance and angle measure, all corresponding sides and angles are congruent. That means that one way to decide whether a pair of triangles are congruent would be to measure all of the sides and angles.

Contents

- 1 How do you prove SAS congruence?
- 2 Why can a sequence of rigid transformations result in congruent figures?
- 3 How do you use rigid motions to verify congruence?
- 4 What rigid motion is represented with the congruent triangles?
- 5 Which rigid motion could be used to prove alternate interior angles are congruent?
- 6 Are rigid transformations similar or congruent?
- 7 What does a rigid motion transformation preserve?
- 8 What is an example of a rigid transformation?
- 9 Do rigid transformations preserve distance?

### How are rigid transformations used?

Rigid transformations, like rotations and reflections, change a shape’s position but keep its size and shape. These transformations preserve side lengths, angle measures, perimeter, and area. But they might not keep the same coordinates or relationships to lines outside the figure.

## How do you prove SAS congruence?

Side-Angle-Side (SAS) Congruence Theorem If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

## Why can a sequence of rigid transformations result in congruent figures?

Recall that rigid transformations preserve distance and angles. This means that congruent figures will have corresponding angles and sides that are the same measure and length.

#### What are three rigid motions that can be used to prove triangle congruence?

If the triangles are congruent, you can use a sequence of translations, reflections, and rotations to map one triangle onto the other. Sequence of rigid motions will map one triangle to another.

## How do you use rigid motions to verify congruence?

A method used to determine whether two segments or angles are congruent is to find and compare their measures. Additionally, their congruence can be determined through the use of rigid motions. Engage with this lesson to learn how this is done. In the following applet, four figures are shown. When working with the different types of rigid motions, it was used that they map a segment into another segment. To verify that this is true, consider a segment and any rigid motion. Let be the image of and the image of Because rigid motions preserve distances, is equal to Now, to check that every point of was actually mapped onto consider a point on different from the endpoints. Let be the image of under the rigid motion. The idea is to show that lies on between and To do this, it must be checked that is equal to Again, since rigid motions preserve distances, and Next, Equations (I) and (II) can be added together and simplified using the Segment Addition Postulate, Segment Addition Postulate Consequently, lies on which proves that the image of every point of lies on Therefore, rigid motions map segments into segments. Not only can rigid motions map segments into segments, but rigid motions also map angles into congruent angles, Because of these two properties, to find only the image of the vertices provides enough information to map the image of a polygon under a rigid motion. a If the center and the radius of are known, how can the image be found? b If the center and a point on are given, how can the image be found? c If three points on are given, how can the image be found? a The image of is the circle centered at the image of the center of and the same radius as b The image of is the circle centered at the image of the center of that passes through the image of the given point.

c The image of is the circle that passes through the images of the three given points. a Use the fact that rigid motions preserve distances. b With the image of the center and the image of the given point, the image circle can be drawn. c Consider the segments connecting the images of the given points.

Then, draw their perpendicular bisectors, These lines intersect at the center of the circle. a The first step is to find the image of the center of under the rigid motion. Since rigid motions preserve distances, the image of is a circle whose radius is equal to the radius of Therefore, is the circle centered at and radius b As in Part A, start by finding the image of the center of and the image of the given point. The image of is a circle centered at with radius Since and are equal, is the circle centered at passing through point c Once more, start by finding the images of the three given points. The image of is a circle passing through and Next, find the center of by drawing the perpendicular bisectors of and The point of intersection of the perpendicular bisectors is the center of That way, the image of can be drawn. Since rigid motions map segments into congruent segments and angles into congruent angles, rigid motions can be used to define when a pair of geometric figures are congruent. Two figures are congruent figures if there is a rigid motion or sequence of rigid motions that maps one of the figures onto the other. When writing a polygon congruence, the corresponding vertices must be listed in the same order. For the polygons above, two of the possible congruence statements can be written as follows. Emily and her family are spending the weekend at a lake house. On Saturday afternoon, Emily begins to work on her geometry homework that asks her to determine if the following pair of polygons are congruent, Unfortunately, Emily left her ruler and protractor at her house and only brought a pencil and a piece of tracing paper. a How can Emily do the homework? b Are the polygons congruent? a Emily can use the definition of congruent figures in terms of rigid motions.

She can draw one of the polygons on the tracing paper and use it to perform different rigid motions on it and try to match both polygons. b Yes, by applying a sequence of rigid motions, one polygon can be mapped onto the other. b After applying some rigid motions, if the two figures match perfectly, then they are congruent.

a Since Emily has no ruler and protractor, she cannot compare the figures’ side lengths or angle measures. However, she can check if the polygons are congruent using rigid motions. By using paper and pencil she can perform rigid motions on the figures. Then, draw one of the figures on tracing paper and translate it so that the corresponding vertices match each other. Since the rest of the vertices do not match, a single translation is not enough to map one figure onto the other. The next step is to try to match a pair of corresponding sides. To do so, rotate the paper about the matching vertices. As seen, the rest of the sides still do not match. Therefore, another transformation is needed. Notice that the two given polygons have opposite orientations, which suggests that performing a reflection is helpful. In this case, the line of reflection is the line containing the matching sides. This time, all sides of both figures match. b Since the two polygons matched perfectly and only rigid motions were applied, it can be concluded that the polygons are congruent. When two figures are not congruent, no sequence of rigid motions maps one figure onto the other. A translation can map onto However, the remaining parts of do not match the parts of Equivalently, can be mapped onto by a translation. Still, the remaining parts do not match as before. In the first case described before, could be rotated about in order to map onto However, by doing this, will no longer match with Although the quadrilaterals have the same side lengths, no matter what rigid motion is applied, will not match The reason is that the quadrilaterals have different shapes. Like the shapes of figures, their sizes matter for the figures to be congruent. The next pair of squares have the same shape, but they are not congruent. Consequently, both size and shape are essential when determining if two figures are congruent. Consider the following pair of quadrilaterals and in the coordinate plane, Furthermore, consider the following sequences of transformations,

Translation along followed by a clockwise rotation about the origin, Glide reflection defined by the line and rotation about the origin followed by a reflection across

Are the given quadrilaterals congruent ? In the affirmative case, which of the listed sequences maps onto Use the orientation of both quadrilaterals to discard one of the sequences. Then, apply the remaining sequences to one at a time, to find the image, Compare the image to To determine if any of the given sequences maps onto apply them to one at a time, and compare each resulting image to

### Why are rigid transformations important?

Rigid transformations keep the shape’s size and angles the same. The image is the shape in its new position and direction.

#### Is SAS enough to prove congruence?

4. AAS (angle, angle, side) –

- AAS stands for “angle, angle, side” and means that we have two triangles where we know two angles and the non-included side are equal.
- For example:

is congruent to: |

See to find out more) If two angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

#### Does SAS always prove congruence?

SAS (Side-Angle-Side) – A second way to prove the congruence of triangles is to show that two sides and their included angle are congruent. This method is called side-angle-side. It is important to remember that the angle must be the included angle-otherwise you can’t be sure of congruence. Figure %: Two sides and their included angle determine a triangle

### What is an example of a SAS proof?

Example 1. When two triangles have two pairs of sides and their included angles congruent, the triangles are congruent. What if the angles aren’t included angles? Even though these triangles have two pairs of sides and one pair of angles that are congruent, the triangles are clearly not congruent.

### What transformations prove congruence?

There are three main types of congruence transformations: reflections (flips), rotations (turns), and translations (slides). These congruence transformations can be used to obtain congruent shapes or to verify that two shapes are congruent.

Congruence transformations Using three forms of transformations, Rotations, Reflections and Translations, we can create congruent shapes. In fact all pairs of congruent shapes can be matched to each other using a series or one or more of these three transformations.

### Are rigid transformations always congruent?

Rigid transformations create congruent figures. You might think of congruent figures as shapes that ‘look exactly the same’, but congruent figures can always be linked to rigid transformations as well.

### What are the properties of the 3 rigid transformations?

There are three main types of rigid transformations. These are rotations, reflections, and translations. Each of rotations, reflections, and translations will preserve the distances between each pair of points of the object, and they will preserve the overall shape and size of the object.

## What rigid motion is represented with the congruent triangles?

Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Possible Answers: Any way of moving a figure such that the relative distance between the points/vertices of the figure stay the same and the relative position of the points/vertices of the figure stay the same Any way of moving a figure Any way of moving a figure such that the relative position of the points/vertices of the figure stay the same but the distance between points/vertices can differ Any way of moving a figure such that the relative distance between the points/vertices of the figure stay the same but the position can differ Correct answer: Any way of moving a figure such that the relative distance between the points/vertices of the figure stay the same and the relative position of the points/vertices of the figure stay the same Explanation : Rigid motion follows these criteria because the motion is rigid meaning that everything that is moving stays the same except for the location of the entire figure.

- There are three common types of rigid motion; translation, reflection, and rotation.
- In terms of rigid motion, how do we know when two figures are congruent to one another? Possible Answers: Two figures are congruent if they meet the criteria of one of the following theorems: SAS, ASA, SSS Two figures are congruent if they meet the criteria of all three of the following theorems: SAS, ASA, SSS Two figures are congruent if there is a sequence of rigid motions that maps one figure to another Two figures are congruent if there is a sequence of rigid motions that maps at least two vertices to another Correct answer: Two figures are congruent if there is a sequence of rigid motions that maps one figure to another Explanation : This is the correct definition in terms of rigid motions.

Some of the other options are correct definitions for congruence but do not mention the criteria of there being rigid motion between the two figures. An example of this is that and are congruent because they are a reflection of one another. Their vertices that map to each other are The following two triangles are congruent by the SAS Theorem. What are the series of rigid motions that map them to one another? (Figures not to scale) Possible Answers: Translation, rotation, reflection Correct answer: Translation, rotation, reflection Explanation :

- Now they share a vertex and we are able to rotate them together mapping to and to,
- Now we can reflect across mapping to, to, and to,
- So the order of the series of rigid motions is translation, rotation, reflection.

True or False: If two triangles are congruent through SAS Theorem and share a vertex, they will follow the rigid motions of rotation and reflection. Possible Answers: Explanation : Consider the triangles and, where, We are able to rotate them together mapping to and to,

- Now we can reflect across mapping ro, and to,
- Now we are left with the two congruent triangles lying on top of one another, proving that the rigid motions that map these two triangles to one another are rotation and reflection.

Triangles that share a side and follow the SSS criteria for congruence follow which of the following rigid motions? Possible Answers: None of the answer choices are correct Correct answer: Reflection Explanation : Consider the following triangles, and,

- They share the side,
- If we reflect triangle across, we match up all congruent sides, mapping them to one another and mapping to, to, and to, proving these two triangles congruent.
- True or False: The following triangles are congruent by two different methods: Possible Answers: Explanation : Let’s first begin by showing that these two triangles are congruent through a series of rigid motions.

Let’s use our point of reference be and since we know that these angles are congruent through the information given in the picture. We are able to map to by reflecting along the line, So these two triangles are congruent. Now we will show that these two triangles are congruent through another theorem.

- We see that there are two pairs of corresponding congruent angles,, and the angles’ included sides are congruent as well,,
- The ASA Theorem states that if two triangles share two pairs of corresponding congruent angles and their included sides are congruent, then these two triangles are congruent.
- So by ASA, these triangles are congruent.

Tell why the following triangles are congruent both by rigid motions and one of the three triangle congruence theorems. Possible Answers: Correct answer: SSS, reflection Explanation : We can see that, We know that by reflexive property. So by the SSS Theorem, these two triangles are congruent.

- We can also reflect triangle across line to map the remaining angles to one another; to,
- So these triangles are proven congruent through reflection as well.
- Through which rigid motion are the following triangles related by? Possible Answers: None of the choices are correct Correct answer: Reflection Explanation : This becomes more clear with the orange line between the two triangles.

If flipped over this orange line, the two figures would match up their corresponding congruent parts creating the same triangle. Give an informal proof that proves the following two triangles are congruent by the SAS Theorem and by a series of rigid motions.

- Possible Answers: The SAS Theorem states that if two triangles share two pairs of corresponding congruent sides are congruent and their included angle is also congruent, then these two triangles are congruent.
- We are given that and so these two triangles are congruent by the SAS theorem.
- We are able to map to by rotating 180 degrees clockwise.

So these two triangles are congruent by rigid motion. The SAS Theorem states that if two triangles share two pairs of corresponding congruent sides are congruent and their included angle is also congruent, then these two triangles are congruent. We are given that and so these two triangles are congruent by the ASA theorem.

We are able to map to by reflecting, So these two triangles are congruent by rigid motion. The SAS Theorem states that if two triangles share two pairs of corresponding congruent sides are congruent and their included angle is also congruent, then these two triangles are congruent. We are given that and so these two triangles are congruent by the ASA theorem.

We are able to map to by translating, So these two triangles are congruent by rigid motion. Correct answer: The SAS Theorem states that if two triangles share two pairs of corresponding congruent sides are congruent and their included angle is also congruent, then these two triangles are congruent.

- We are given that and so these two triangles are congruent by the SAS theorem.
- We are able to map to by rotating 180 degrees clockwise.
- So these two triangles are congruent by rigid motion.
- Explanation : The SAS Theorem states that if two triangles share two pairs of corresponding congruent sides are congruent and their included angle is also congruent, then these two triangles are congruent.

We are given that and so these two triangles are congruent by the SAS theorem. We are able to map to by rotating 180 degrees clockwise. So these two triangles are congruent by rigid motion. The following two triangles are congruent by the ASA Theorem. What are the series of rigid motions that map them to one another? Possible Answers: Correct answer: Rotation, reflection Explanation : First, the two triangles and share a vertex, so we know that maps to by the reflective property.

- Now we can reflect the triangle across to map to, to and to,
- So the order of rigid motions is rotation, reflection.

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Charles Cohn Varsity Tutors LLC 101 S. Hanley Rd, Suite 300 St. Louis, MO 63105 Or fill out the form below: : Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

## Which rigid motion could be used to prove alternate interior angles are congruent?

proving alternate interior angles congruent using rigid motions – Easing the Hurry Syndrome CCSS-M.G-CO.C.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent ; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

We make sense of (wording below from Cut the Knot):If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles.We use dynamic geometry software to explore :

And then traditionally, we have allowed corresponding angles congruent when parallel lines are cut by a transversal as the postulate in our deductive system. It makes sense to students that the corresponding angles are congruent. Then once we’ve allowed those, it’s not too bad to prove that alternate interior angles are congruent when parallel lines are cut by a transversal.

But we wonder whether we have to let corresponding angles in as a postulate. Can we use rigid motions to show that the corresponding angles are congruent? One student suggested constructing the midpoint, X, of segment BE. Then we created a parallel to lines m and n through X. That didn’t get us very far in showing that the corresponding angles are congruent.

(image on the top left) Another student suggested translating line m using vector BE. So we really translated more than just line m, We really translated the upper half-plan formed by line m, We used took a picture of the top part of the diagram (line m and above) and translated it using vector BE.

- We can see in the picture on the right, that m maps to n and the transversal maps to itself, and so we conclude (bottom left image) that ∠CBA is congruent to ∠DEB: if two parallel lines are cut by a transversal, the corresponding angles are congruent.
- Once corresponding angles are congruent, then proving alternate interior (or exterior) angles congruent or consecutive interior (or exterior) angles supplementary when two parallel lines are cut by a transversal follows using a mix of congruent vertical angles, transitive and/or substitution, Congruent Supplements.

But can we prove that alternate interior angles are congruent when parallel lines are cut by a transversal using rigid motions? Several students suggested we could do the same translation (translating the “top” parallel line onto the “bottom” parallel line).

- 2≅∠2′ because of the translation (and because they are corresponding), and we can say that ∠2’≅∠3 since we have already proved that vertical angles are congruent.
- 2≅∠3 using the Transitive Property of Congruence.
- We conclude that when two parallel lines are cut by a transversal, alternate interior angles are congruent.

Another team suggested constructing the midpoint M of segment XY (top image). They rotated the given lines and transversal 180˚ about M (bottom image). ∠2 has been carried onto ∠3 and ∠3 has been carried onto ∠2. We conclude that when two parallel lines are cut by a transversal, alternate interior angles are congruent.

Another team constructed the same midpoint as above with a line parallel to the given lines through that midpoint. They reflected the entire diagram about that line, which created the line in red. They used the base angles of an isosceles triangle to show that alternate interior angles are congruent. Note 1: We are still postulating that through a point not on a line there is exactly one line parallel to the given line.

This is what textbooks I’ve used in the past have called the parallel postulate, And we are postulating that the distance between parallel lines is constant. Note 2: We haven’t actually proven that the base angles of an isosceles triangle are congruent.

But students definitely know it to be true from their work in middle school. The proof is coming soon. Note 3: Many of these same ideas will show that consecutive (or same-side) interior angles are supplementary. We can use rigid motions to make the images of two consecutive interior angles form a linear pair.

After the lesson, a colleague suggested an Illustrative Mathematics task on, which helped me think through the validity of the arguments that my students made. As the journey continues, I find the tasks, commentary, and solutions on IM to be my own textbook – a dynamic resource for learners young and old.

### Is congruence a term used with rigid motion?

Two figures are congruent if there is a rigid motion carrying one onto the other—this is the “principle of superposition” according to the Common Core. This definition of congruence agrees with our intuition that congruent figures should have the same size and shape. Orientation, location, and rotation don’t matter!.

## Are rigid transformations similar or congruent?

If we can map one figure onto another using rigid transformations, they are congruent. They are still congruent if we need to use more than one transformation to map it. They aren’t if we use a transformation that changes the size of the shape.

### What is a rigid motion and why is it important?

In this section we will learn about isometry or rigid motions. An isometry is a transformation that preserves the distances between the vertices of a shape. A rigid motion does not affect the overall shape of an object but moves an object from a starting location to an ending location. The resultant figure is congruent to the original figure.

A rigid motion is when an object is moved from one location to another and the size and shape of the object have not changed. |

table>

Two figures are congruent if and only if there exists a rigid motion that sets up a correspondence of one figure as the image of the other. Side lengths remain the same and interior angles remain the same.table>

An identity motion is a rigid motion that moves an object from its starting location to exactly the same location. It is as if the object has not moved at all.There are four kinds of rigid motions: translations, rotations, reflections, and glide-reflections. When describing a rigid motion, we will use points like P and Q, located on the geometric shape, and identify their new location on the moved geometric shape by P’ and Q’., Figure \(\PageIndex \): Translation The translation of the blue triangle with point P was moved along the vector to the location of the red triangle with point P’. Also note that the other vertices of the blue triangle also moved along the vector to corresponding vertices on the red triangle. P’ P

A translation of an object moves the object along a directed line segment called a vector for a specific distance and in a specific direction. The motion is completely determined by two points P and P’ where P is on the original object and P’ is on the translated object. |

In regular language, a translation of an object is a slide from one position to another. You are given a geometric figure and an arrow which represents the vector. The vector gives you the direction and distance which you slide the figure. Example \(\PageIndex \) Translation of a Triangle You are given a blue triangle and a vector, Move the triangle along vector, Figure \(\PageIndex \): Blue Triangle and Vector

B | ||||||||||

A | C | |||||||||

Figure \(\PageIndex \): Result of the Translation

B’ | ||||||||||

A’ | C’ | |||||||||

B | ||||||||||

A | C | |||||||||

Properties of a Translation

A translation is completely determined by two points P and P’ Has no fixed points Has identity motion

Note: the vector has the same length as vector, but points in the opposite direction. Example \(\PageIndex \) Translation of an Object Given the L-shape figure below, translate the figure along the vector, The vector moves horizontally three units to the right and vertically two units up. Move each vertex three units to the right and two units up. The red figure is the position of the L-shape figure after the slide. Figure \(\PageIndex \): L-Shape and Vector

P | ||||||||||||||

Figure \(\PageIndex \): Result of the L-Shape Translated by Vector

P’ | ||||||||||||||

P | ||||||||||||||

The next type of transformation (rigid motion) that we will discuss is called a rotation. A rotation moves an object about a fixed point R called the rotocenter and through a specific angle. The blue triangle below has been rotated 90° about point R.

A rotation of an object moves the object around a point called the rotocenter R a certain angle either clockwise or counterclockwise. |

Note: the rotocenter R can be outside the object, inside the object or on the object. Figure \(\PageIndex \): A Triangle Rotated 90° around the Rotocenter R outside the Triangle 90° R Figure \(\PageIndex \): A Triangle Rotated 180° around the Rotocenter R inside the Triangle R Properties of a Rotation

A Rotation is completely determined by two pairs of points; P and P’ and

Q and Q’

Has one fixed point, the rotocenter R Has identity motion the 360° rotation

Example \(\PageIndex \): Rotation of an L-Shape Given the diagram below, rotate the L-shaped figure 90° clockwise about the rotocenter R. The point Q rotates 90°. Move each vertex 90° clockwise. Figure \(\PageIndex \): L-Shape and Rotocenter R The L-shaped figure will be rotated 90° clockwise and vertex Q will move to vertex Q’. Each vertex of the object will be rotated 90°.

Q | 90° | Q’ | ||||||||||

R | ||||||||||||

Figure \(\PageIndex \): Result of the 90° Clockwise Rotation

Q | Q’ | |||||||||||

R | ||||||||||||

Example \(\PageIndex \): 45° Clockwise Rotation of a Rectangle Figure \(\PageIndex \): Rectangle and Rotocenter R

Q | 45° | |||||||||||

Q’ | ||||||||||||

R | ||||||||||||

Figure \(\PageIndex \): Result of 45° Clockwise Rotation

Q | ||||||||||||

Q’ | ||||||||||||

R | ||||||||||||

Example \(\PageIndex \): 180° Clockwise Rotation of an L-Shape Figure \(\PageIndex \): L-Shape and Rotocenter R

A | |||||||||||

B | R | 180° | |||||||||

Figure \(\PageIndex \): Result of the 180° Clockwise Rotation

A | |||||||||||

B | R | B’ | |||||||||

A’ |

The next type of transformation (rigid motion) is called a reflection. A reflection is a mirror image of an object, or can be thought of as “flipping” an object over.

Reflection: If each point on a line corresponds to itself, and each other point in the plane corresponds to a unique point in the plane, such that is the perpendicular bisector of, then the correspondence is called the reflection in line, |

In regular language, a reflection is a mirror image across a line, The line is the midpoint of the line between the two points, P in the original figure and P’ in the reflection. P goes to P’. Figure \(\PageIndex \): Reflection of an Object about a Line l

C | ||||||||||||

B | ||||||||||||

A | ||||||||||||

l | ||||||||||||

Figure \(\PageIndex \): Result of the Reflection over Line l The reflection places each vertex along a line perpendicular to l and equidistant from l.

C’ | C | |||||||||||

B’ | B | |||||||||||

A’ | A | |||||||||||

l | ||||||||||||

Properties of a Reflection

A reflection is completely determined by a single pair of points; P and P’ Has infinitely many fixed points: the line of reflection l Has identity motion the reverse reflection

Example \(\PageIndex \) Reflect an L-Shape across a Line l Figure \(\PageIndex \): L-shape and Line l

B | ||||||||||||

C | ||||||||||||

A | ||||||||||||

l |

Reflect the L-shape across line l. The red L-shape shown below is the result after the reflection. The original position of each vertex is on a line with the reflected position of each vertex. This line that connects the original and reflected positions of the vertex is perpendicular to line l and the original and reflected positions of each vertex are equidistant to line l,

B’ | ||||||||||||

C’ | ||||||||||||

l | A’ | |||||||||||

Example \(\PageIndex \): Reflect another L-Shape across Line l First identify the vertices of the figure. From each vertex, draw a line segment perpendicular to line l and make sure its midpoint lies on line l. Now draw the new positions of the vertices, making the transformed figure a mirror image of the original figure. Figure \(\PageIndex \): L-Shape and Line l

B | ||||||||||

A | l | |||||||||

C | ||||||||||

D | ||||||||||

Figure \(\PageIndex \): Result of Reflection over Line l

B | ||||||||||

A | ||||||||||

C | ||||||||||

D | C’ | B’ | ||||||||

A’ | ||||||||||

D’ | ||||||||||

The final transformation (rigid motion) that we will study is a glide-reflection, which is simply a combination of two of the other rigid motions.

A glide-reflection is a combination of a reflection and a translation. |

Example \(\PageIndex \) Glide-Reflection of a Smiley Face by Vector and Line l Figure \(\PageIndex \): Smiley Face, Vector, and Line l

l | ||||||||||

Figure \(\PageIndex \): Smiley Face Glide-Reflection Step One First slide the smiley face two units to the right along the vector,

l | ||||||||||

Figure \(\PageIndex \): Smiley Face Glide-Reflection Step Two Then reflect the smiley face across line l. The final result is the green upside-down smiley face.

l | ||||||||||

Properties of a Glide-Reflection

A reflection is completely determined by a single pair of points; P and P. Has infinitely fixed points: the line of reflection l. Has identity motion the reverse glide-reflection,

Example \(\PageIndex \): Glide-Reflection of a Blue Triangle Figure \(\PageIndex \): Blue Triangle, Vector, and Line l

l | ||||||||||

Figure \(\PageIndex \): Triangle Glide-Reflection Step One First, slide the triangle along vector,

l | ||||||||||

P* | ||||||||||

P | ||||||||||

Figure \(\PageIndex \): Triangle Glide-Reflection Step Two Then, reflect the triangle across line l. The final result is the green triangle below line l,

Q* | ||||||||||

P* | ||||||||||

S* | ||||||||||

P’ | ||||||||||

S’ | Q’ | |||||||||

Example \(\PageIndex \): Glide-Reflection of an L-Shape Figure \(\PageIndex \): L-Shape, Vector, and Line l

l | ||||||||||

Figure \(\PageIndex \): L-Shape Glide-Reflection Step One First slide the L-shape along vector,

B* | ||||||||||

B | A* | |||||||||

A | ||||||||||

Figure \(\PageIndex \): L-Shape Glide-Reflection Step Two Then reflect the L-shape across line l. The result is the green open shape below the line l,

B* | ||||||||||

A’ | ||||||||||

B | ||||||||||

B’ | ||||||||||

## What does a rigid motion transformation preserve?

Rigid motions preserve collinearity. Reflections, rotations, and translations are all rigid motions. So, they all preserve distance, angle measure, betweenness, and collinearity.

#### What is a rigid motion and why is it important?

In this section we will learn about isometry or rigid motions. An isometry is a transformation that preserves the distances between the vertices of a shape. A rigid motion does not affect the overall shape of an object but moves an object from a starting location to an ending location. The resultant figure is congruent to the original figure.

A rigid motion is when an object is moved from one location to another and the size and shape of the object have not changed. |

table>

Two figures are congruent if and only if there exists a rigid motion that sets up a correspondence of one figure as the image of the other. Side lengths remain the same and interior angles remain the same.table>

An identity motion is a rigid motion that moves an object from its starting location to exactly the same location. It is as if the object has not moved at all.There are four kinds of rigid motions: translations, rotations, reflections, and glide-reflections. When describing a rigid motion, we will use points like P and Q, located on the geometric shape, and identify their new location on the moved geometric shape by P’ and Q’., Figure \(\PageIndex \): Translation The translation of the blue triangle with point P was moved along the vector to the location of the red triangle with point P’. Also note that the other vertices of the blue triangle also moved along the vector to corresponding vertices on the red triangle. P’ P

A translation of an object moves the object along a directed line segment called a vector for a specific distance and in a specific direction. The motion is completely determined by two points P and P’ where P is on the original object and P’ is on the translated object. |

In regular language, a translation of an object is a slide from one position to another. You are given a geometric figure and an arrow which represents the vector. The vector gives you the direction and distance which you slide the figure. Example \(\PageIndex \) Translation of a Triangle You are given a blue triangle and a vector, Move the triangle along vector, Figure \(\PageIndex \): Blue Triangle and Vector

B | ||||||||||

A | C | |||||||||

Figure \(\PageIndex \): Result of the Translation

B’ | ||||||||||

A’ | C’ | |||||||||

B | ||||||||||

A | C | |||||||||

Properties of a Translation

A translation is completely determined by two points P and P’ Has no fixed points Has identity motion

Note: the vector has the same length as vector, but points in the opposite direction. Example \(\PageIndex \) Translation of an Object Given the L-shape figure below, translate the figure along the vector, The vector moves horizontally three units to the right and vertically two units up. Move each vertex three units to the right and two units up. The red figure is the position of the L-shape figure after the slide. Figure \(\PageIndex \): L-Shape and Vector

P | ||||||||||||||

Figure \(\PageIndex \): Result of the L-Shape Translated by Vector

P’ | ||||||||||||||

P | ||||||||||||||

The next type of transformation (rigid motion) that we will discuss is called a rotation. A rotation moves an object about a fixed point R called the rotocenter and through a specific angle. The blue triangle below has been rotated 90° about point R.

A rotation of an object moves the object around a point called the rotocenter R a certain angle either clockwise or counterclockwise. |

Note: the rotocenter R can be outside the object, inside the object or on the object. Figure \(\PageIndex \): A Triangle Rotated 90° around the Rotocenter R outside the Triangle 90° R Figure \(\PageIndex \): A Triangle Rotated 180° around the Rotocenter R inside the Triangle R Properties of a Rotation

A Rotation is completely determined by two pairs of points; P and P’ and

Q and Q’

Has one fixed point, the rotocenter R Has identity motion the 360° rotation

Example \(\PageIndex \): Rotation of an L-Shape Given the diagram below, rotate the L-shaped figure 90° clockwise about the rotocenter R. The point Q rotates 90°. Move each vertex 90° clockwise. Figure \(\PageIndex \): L-Shape and Rotocenter R The L-shaped figure will be rotated 90° clockwise and vertex Q will move to vertex Q’. Each vertex of the object will be rotated 90°.

Q | 90° | Q’ | ||||||||||

R | ||||||||||||

Figure \(\PageIndex \): Result of the 90° Clockwise Rotation

Q | Q’ | |||||||||||

R | ||||||||||||

Example \(\PageIndex \): 45° Clockwise Rotation of a Rectangle Figure \(\PageIndex \): Rectangle and Rotocenter R

Q | 45° | |||||||||||

Q’ | ||||||||||||

R | ||||||||||||

Figure \(\PageIndex \): Result of 45° Clockwise Rotation

Q | ||||||||||||

Q’ | ||||||||||||

R | ||||||||||||

Example \(\PageIndex \): 180° Clockwise Rotation of an L-Shape Figure \(\PageIndex \): L-Shape and Rotocenter R

A | |||||||||||

B | R | 180° | |||||||||

Figure \(\PageIndex \): Result of the 180° Clockwise Rotation

A | |||||||||||

B | R | B’ | |||||||||

A’ |

The next type of transformation (rigid motion) is called a reflection. A reflection is a mirror image of an object, or can be thought of as “flipping” an object over.

Reflection: If each point on a line corresponds to itself, and each other point in the plane corresponds to a unique point in the plane, such that is the perpendicular bisector of, then the correspondence is called the reflection in line, |

In regular language, a reflection is a mirror image across a line, The line is the midpoint of the line between the two points, P in the original figure and P’ in the reflection. P goes to P’. Figure \(\PageIndex \): Reflection of an Object about a Line l

C | ||||||||||||

B | ||||||||||||

A | ||||||||||||

l | ||||||||||||

Figure \(\PageIndex \): Result of the Reflection over Line l The reflection places each vertex along a line perpendicular to l and equidistant from l.

C’ | C | |||||||||||

B’ | B | |||||||||||

A’ | A | |||||||||||

l | ||||||||||||

Properties of a Reflection

A reflection is completely determined by a single pair of points; P and P’ Has infinitely many fixed points: the line of reflection l Has identity motion the reverse reflection

Example \(\PageIndex \) Reflect an L-Shape across a Line l Figure \(\PageIndex \): L-shape and Line l

B | ||||||||||||

C | ||||||||||||

A | ||||||||||||

l |

Reflect the L-shape across line l. The red L-shape shown below is the result after the reflection. The original position of each vertex is on a line with the reflected position of each vertex. This line that connects the original and reflected positions of the vertex is perpendicular to line l and the original and reflected positions of each vertex are equidistant to line l,

B’ | ||||||||||||

C’ | ||||||||||||

l | A’ | |||||||||||

Example \(\PageIndex \): Reflect another L-Shape across Line l First identify the vertices of the figure. From each vertex, draw a line segment perpendicular to line l and make sure its midpoint lies on line l. Now draw the new positions of the vertices, making the transformed figure a mirror image of the original figure. Figure \(\PageIndex \): L-Shape and Line l

B | ||||||||||

A | l | |||||||||

C | ||||||||||

D | ||||||||||

Figure \(\PageIndex \): Result of Reflection over Line l

B | ||||||||||

A | ||||||||||

C | ||||||||||

D | C’ | B’ | ||||||||

A’ | ||||||||||

D’ | ||||||||||

The final transformation (rigid motion) that we will study is a glide-reflection, which is simply a combination of two of the other rigid motions.

A glide-reflection is a combination of a reflection and a translation. |

Example \(\PageIndex \) Glide-Reflection of a Smiley Face by Vector and Line l Figure \(\PageIndex \): Smiley Face, Vector, and Line l

l | ||||||||||

Figure \(\PageIndex \): Smiley Face Glide-Reflection Step One First slide the smiley face two units to the right along the vector,

l | ||||||||||

Figure \(\PageIndex \): Smiley Face Glide-Reflection Step Two Then reflect the smiley face across line l. The final result is the green upside-down smiley face.

l | ||||||||||

Properties of a Glide-Reflection

A reflection is completely determined by a single pair of points; P and P. Has infinitely fixed points: the line of reflection l. Has identity motion the reverse glide-reflection,

Example \(\PageIndex \): Glide-Reflection of a Blue Triangle Figure \(\PageIndex \): Blue Triangle, Vector, and Line l

l | ||||||||||

Figure \(\PageIndex \): Triangle Glide-Reflection Step One First, slide the triangle along vector,

l | ||||||||||

P* | ||||||||||

P | ||||||||||

Figure \(\PageIndex \): Triangle Glide-Reflection Step Two Then, reflect the triangle across line l. The final result is the green triangle below line l,

Q* | ||||||||||

P* | ||||||||||

S* | ||||||||||

P’ | ||||||||||

S’ | Q’ | |||||||||

Example \(\PageIndex \): Glide-Reflection of an L-Shape Figure \(\PageIndex \): L-Shape, Vector, and Line l

l | ||||||||||

Figure \(\PageIndex \): L-Shape Glide-Reflection Step One First slide the L-shape along vector,

B* | ||||||||||

B | A* | |||||||||

A | ||||||||||

Figure \(\PageIndex \): L-Shape Glide-Reflection Step Two Then reflect the L-shape across line l. The result is the green open shape below the line l,

B* | ||||||||||

A’ | ||||||||||

B | ||||||||||

B’ | ||||||||||

## What is an example of a rigid transformation?

Rigid transformation Mathematical transformation that preserves distances

This article may be to readers, In particular, the lead refers correctly to transformations of Euclidean spaces, while the sections describe only the case of Euclidean vector spaces or of spaces of coordinate vectors. The “formal definition” section does not specify which kind of objects are represented by the variables, call them vaguely as “vectors”, suggests implicitly that a basis and a are defined for every kind of vectors. Please help, There might be a discussion about this on, ( August 2021 ) ( ) |

See also: In, a rigid transformation (also called Euclidean transformation or Euclidean isometry ) is a of a that preserves the between every pair of points. The rigid transformations include,,, or any sequence of these. Reflections are sometimes excluded from the definition of a rigid transformation by requiring that the transformation also preserve the of objects in the Euclidean space.

(A reflection would not preserve handedness; for instance, it would transform a left hand into a right hand.) To avoid ambiguity, a transformation that preserves handedness is known as a proper rigid transformation, or rototranslation, Any proper rigid transformation can be decomposed into a rotation followed by a translation, while any improper rigid transformation can be decomposed into an followed by a translation, or into a sequence of reflections.

Any object will keep the same and size after a proper rigid transformation. All rigid transformations are examples of, The set of all (proper and improper) rigid transformations is a called the, denoted E( n ) for n -dimensional Euclidean spaces. The set of proper rigid transformations is called special Euclidean group, denoted SE( n ),

#### What does a rigid motion transformation preserve?

Rigid motions preserve collinearity. Reflections, rotations, and translations are all rigid motions. So, they all preserve distance, angle measure, betweenness, and collinearity.

## Do rigid transformations preserve distance?

Rigid transformations preserve distance and angles. All corresponding sides will be the same length and all corresponding angles will be the same measure.