basic rigid motion proofs common core geometry homework answers

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Common Core: High School - Geometry : Transformation and Congruence of Rigid Motions: CCSS.Math.Content.HSG-CO.B.6

Study concepts, example questions & explanations for common core: high school - geometry, all common core: high school - geometry resources, example questions, example question #1 : transformation and congruence of rigid motions: ccss.math.content.hsg co.b.6.

Determine whether the statement is true or false.

For a translation to be considered rigid, the starting and ending figures must be congruent.

Recall that a rigid motion is that that preserves the distances while undergoing a motion in the plane. This is also called an isometry, rigid transformations, or congruence transformations and there are four different types. Therefore, for the translation to be considered "rigid" the two figures must be congruent by definition of a rigid motion.

Therefore, the statement, "For a translation to be considered rigid, the starting and ending figures must be congruent." is true.

Example Question #2 : Transformation And Congruence Of Rigid Motions: Ccss.Math.Content.Hsg Co.B.6

Which of the following is NOT a rigid motion?

• Translation
• Glide Reflection

Recall that a rigid motion is that that preserves the distances while undergoing a motion in the plane. This is also called an isometry, rigid transformations, or congruence transformations and there are four different types.

These basic type of rigid motions include the following:

Therefore, of the answer selections, "Expansion" is the term that is NOT a rigid motion.

None of the other answers

A rigid motion is that that preserves the distances while undergoing a motion in the plane. This is also called an isometry, rigid transformations, or congruence transformations and there are four different types.

Regardless of which type of rigid motion occurred, the following is true about the triangles' angles:

The following is also true about the side lengths:

Example Question #51 : Congruence

Determine whether the statement is true or false:

A glide reflection is a synonym for a rigid motion translation.

A glide reflection is a rigid motion that occurs when a figure is translated a certain distance and then reflected or reflected and then translated. 

A translation is a rigid motion describing when a object is moved a certain distance.

A glide reflection contains a translation but it is not a synonym for translation therefore, the statement is false.

Example Question #6 : Transformation And Congruence Of Rigid Motions: Ccss.Math.Content.Hsg Co.B.6

All glide reflections are reflections.

A glide reflection is a rigid motion that occurs when a figure is translated a certain distance and then reflected or reflected and then translated. In either case, a glide reflection aways contains a reflection.

Therefore, the statement, "All glide reflections are reflections." is true.

Sally has a quarter that is face up on the desk. If she slides it to Bob on the other side of the desk and he flips it over so that the tails side is facing up, is it a rigid motion?

In the situation where Sally has a quarter that is face up on the desk. The coin is the object. Then she slides it to Bob on the other side of the desk and he flips it over so that the tails side is facing up. Therefore, since the coin maintains it shape it is undergoing a translation to reach the other side of the desk and a reflection to flip the coin. This describes a glide reflection which is in fact, a rigid motion.

Therefore, the statement is describing a rigid motion. The answer is "Yes"

Example Question #52 : Congruence

Sally has a quarter that is face up on the desk. Then she slides the coin to Bob on the other side of the desk and he flips it over so that the tails side is facing up. What type of rigid motion does this situation describe?

This is not a rigid motion.

Example Question #53 : Congruence

Jane and Bob are filling up water balloons for a party they are throwing. Jane thinks the balloons should have more water in them so she fills them fuller. Each water balloon's circumference is one inch greater than before. Does this describe a rigid motion?

Recall that a rigid motion is that that preserves the distances between points within the object while undergoing a motion in the plane. This is also called an isometry, rigid transformations, or congruence transformations and there are four different types. The water balloons are the objects in this situation. Since the they are filled with more water to increase their circumferences, it does not preserve the shape of the object and thus, is not a rigid motion. 

Example Question #4 : Transformation And Congruence Of Rigid Motions: Ccss.Math.Content.Hsg Co.B.6

Select the answer that best completes the following definition:

A motion that preserves distance in the plane is called a __________ . 

Transformation

Rigid Motion

Non-Rigid Motion

Therefore, "A motion that preserves distance in the plane is called a rigid motion ".

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10.1: Transformations Using Rigid Motions

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• Page ID 32000

• Maxie Inigo, Jennifer Jameson, Kathryn Kozak, Maya Lanzetta, & Kim Sonier
• Coconino Community College

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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.

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 \(\PageIndex{1}\): Translation

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 \(\PageIndex{1}\) Translation of a Triangle

Figure \(\PageIndex{3}\): Result of the Translation

Properties of a Translation

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

Example \(\PageIndex{2}\) Translation of an Object

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.

Note: the rotocenter R can be outside the object, inside the object or on the object.

Figure \(\PageIndex{6}\): A Triangle Rotated 90° around the Rotocenter R outside the Triangle

Figure \(\PageIndex{7}\): A Triangle Rotated 180° around the Rotocenter R inside the Triangle

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 \(\PageIndex{3}\): 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 \(\PageIndex{8}\): 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°.

Figure \(\PageIndex{9}\): Result of the 90° Clockwise Rotation

Example \(\PageIndex{4}\): 45° Clockwise Rotation of a Rectangle

Figure \(\PageIndex{10}\): Rectangle and Rotocenter R

Figure \(\PageIndex{11}\): Result of 45° Clockwise Rotation

Example \(\PageIndex{5}\): 180° Clockwise Rotation of an L-Shape

Figure \(\PageIndex{12}\): L-Shape and Rotocenter R

Figure \(\PageIndex{13}\): Result of the 180° Clockwise Rotation

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.

Figure \(\PageIndex{14}\): Reflection of an Object about a Line l

Figure \(\PageIndex{15}\): Result of the Reflection over Line l

The reflection places each vertex along a line perpendicular to l and equidistant from 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 \(\PageIndex{6}\) Reflect an L-Shape across a Line l

Figure \(\PageIndex{16}\): L-shape and Line 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 .

Figure \(\PageIndex{17}\): Result of Reflection over Line l

Example \(\PageIndex{7}\): 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 \(\PageIndex{18}\): L-Shape and Line l

Figure \(\PageIndex{19}\): Result of Reflection over Line l

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.

Figure \(\PageIndex{21}\): Smiley Face Glide-Reflection Step One

Figure \(\PageIndex{22}\): Smiley Face Glide-Reflection Step Two

Then reflect the smiley face across line l. The final result is the green upside-down smiley face.

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 \(\PageIndex{9}\): Glide-Reflection of a Blue Triangle

Figure \(\PageIndex{24}\): Triangle Glide-Reflection Step One

Figure \(\PageIndex{25}\): Triangle Glide-Reflection Step Two

Then, reflect the triangle across line l. The final result is the green triangle below line l .

Example \(\PageIndex{10}\): Glide-Reflection of an L-Shape

Figure \(\PageIndex{27}\): L-Shape Glide-Reflection Step One

Figure \(\PageIndex{28}\): 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 .

Common Core State Standards Initiative

High School: Geometry » Congruence

Standards in this domain:, experiment with transformations in the plane, understand congruence in terms of rigid motions, prove geometric theorems, make geometric constructions.

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TRAPS & PITFALLS

The most difficult part of this is the correct ordering of the sides and angles. Students confuse AAS with ASA quite a bit, they struggle to determine if the side is between the angles or not. Students often ask about where to start when naming the criteria… this can be difficult at first to see.  I have them label an S or an A on the diagram and I talk about how you must know the next side or the next consecutive angle to continue naming in that direction… if not.. go the other way.

PAST CONNECTIONS

Students will use the concepts covered in G.CO.4 and G.CO.5. The definitions and the composite transformations will link isometry to congruence.

FUTURE CONNECTIONS

Knowing when two triangles are congruent is essential to so many areas that follow. Triangles being the simplest polygon appear everywhere and their properties and correspondence of congruent parts unlock many geometric mysteries.

MY REFLECTIONS (over line l )

I created an exploration where students create triangle based on certain criteria and then they compare (map) their triangles with a group. This went very well. They quickly could see what things were needed to 'lockdown' the shape. Going from AA to ASA, was a very clear moment for them. They recognized right away that the side held things in check and at least one side was necessary for congruence.

I do discuss the cases of ASS here when teaching an honors class.  It is natural and isn’t too much to bite off.  I find that introducing the ‘swinging side’ opens their eyes to non-congruent cases.  Some time placed here makes teaching the cases of the Law of Sines later much easier because some student already understand the ambiguous case.

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