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7. Rigid Motion and Congruence

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Chapter 1

Rigid Motion and Congruence

Explore the concept of rigid motions and their relationship with congruent figures. Rigid motions, which include transformations like translations, rotations, and reflections, play a pivotal role in understanding the congruence of geometric shapes. When two figures can be mapped onto each other using only rigid motions, they are deemed congruent. This lesson emphasizes the significance of understanding these transformations, especially when determining the congruence of complex geometric figures. Real-world examples, such as determining the congruence of logos or analyzing the properties of various polygons, further illustrate the practical applications of these concepts. The lesson also touches upon the importance of sequence in rigid motions and how different sequences can lead to varied results. Overall, the lesson provides a comprehensive insight into the relationship between rigid motions and congruence in geometry.

Lesson Settings & Tools

	11 Theory slides
	6 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Image Credits

Slide of 11

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.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Rigid motion
Congruent angles
Congruent segments
Rotation
Translation
Reflection
Glide reflection
Coordinate plane

Challenge

Identifying Congruent Figures

In the following applet, four figures are shown. One of the three figures at the bottom is the result of applying a rigid motion or a sequence of rigid motions to the upper figure. Match the preimage with its image.

Discussion

Congruence and Rigid Motions

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 $\overline{A B}$ and any rigid motion. Let $C$ be the image of $A$ and $D$ the image of $B .$

Because rigid motions preserve distances, $C D$ is equal to $A B .$ Now, to check that every point of $\overline{A B}$ was actually mapped onto $\overline{C D},$ consider a point $P$ on $\overline{A B}$ different from the endpoints. Let $Q$ be the image of $P$ under the rigid motion.

The idea is to show that

Q

lies on

\overline{C D}

between

C

and

D .

To do this, it must be checked that

C Q + Q D

is equal to

C D .

Again, since rigid motions preserve distances,

C Q = A P

and

Q D = P B .

{C Q = A P Q D = P B (I) (II)

Next, Equations (I) and (II) can be added together and simplified using the Segment Addition Postulate.

{C Q = A P Q D = P B (I) (II)

▼

Solve by elimination

SysEqnAdd

$(I):$ Add $II$

{C Q + Q D = A P + P B Q D = P B

$(I):$ Segment Addition Postulate

{C Q + Q D = A B Q D = P B

Substitute

$(I):$ $A B = C D$

{C Q + Q D = C D ✓ Q D = P B

Consequently,

Q

lies on

C D

which proves that the image of every point of

\overline{A B}

lies on

\overline{C D} .

Therefore, rigid motions map segments into segments.

Interactive graph where P can be moved along AB and Q is moved accordingly on CD

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.

Rigid motions map $n -$ sided polygons onto $n -$ sided polygons.

Now, consider a circle. The image of a circle is also a circle. However, circles do not have vertices, so how can the image be found? The following example investigates how to handle this situation.

Example

Congruence in Circles

Consider a rigid motion and a circle $C .$

a If the center and the radius of

C

are known, how can the image be found?

b If the center and a point on

C

are given, how can the image be found?

c If three points on

C

are given, how can the image be found?

Answer

a The image of

C

is the circle centered at the image of the center of

C

and the same radius as

C .

b The image of

C

is the circle centered at the image of the center of

C

that passes through the image of the given point.

c The image of

C

is the circle that passes through the images of the three given points.

Hint

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.

Solution

a The first step is to find the image of the center of

C

under the rigid motion.

Since rigid motions preserve distances, the image of $C$ is a circle whose radius is equal to the radius of $C .$ Therefore, $C^{'}$ is the circle centered at $P^{'}$ and radius $r .$

b As in Part A, start by finding the image of the center of

C

and the image of the given point.

The image of $C$ is a circle centered at $P^{'}$ with radius $P Q .$ Since $P Q$ and $P^{'} Q^{'}$ are equal, $C^{'}$ is the circle centered at $P^{'}$ passing through point $Q^{'} .$

c Once more, start by finding the images of the three given points.

The image of

C

is a circle passing through

A^{'},

B^{'},

and

C^{'} .

Next, find the center of

C^{'}

by drawing the perpendicular bisectors of

\overline{A^{'} B^{'}}

and

\overline{B^{'} C^{'}} .

The point of intersection of the perpendicular bisectors is the center of

C^{'} .

That way, the image of

C

can be drawn.

Discussion

Using Rigid Motions to Determine Congruence

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.

Concept

Congruent Figures

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. As a result, congruent figures have the same size and shape. To denote algebraically that two figures are congruent, the symbol

≅

is used.

Showing that two figures are congruent by mapping one onto the other

When stating that two polygons are congruent, 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.

A B C D E ≅ J K L M N or C D E A B ≅ L M N J K

Example

Determining if Figures are Congruent

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?

Answer

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.

Hint

a Use the definition of congruent figures in terms of rigid motions, and remember how to perform rigid motions using paper. Start by translating one polygon in order to match a pair of corresponding vertices.

b After applying some rigid motions, if the two figures match perfectly, then they are congruent.

Solution

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. The first step is to identify possible corresponding parts.

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.

Discussion

Figures That Are Not Congruent

When two figures are not congruent, no sequence of rigid motions maps one figure onto the other. Even so, it is possible that a rigid motion maps certain parts of the preimage onto their corresponding parts, but not all. For example, consider the following pair of quadrilaterals.

A rectangle and and parallelogram with corresponding congruent sides

A translation can map

\overline{A B}

onto

\overline{P Q} .

However, the remaining parts of

A B C D

do not match the parts of

P Q R S .

Equivalently,

\overline{C D}

can be mapped onto

\overline{R S}

by a translation. Still, the remaining parts do not match as before.

Translating one rectangle so that a pair of side match

In the first case described before,

A^{'} B^{'} C^{'} D^{'}

could be rotated about

B^{'}

in order to map

\overline{B^{'} C^{'}}

onto

\overline{Q R} .

However, by doing this,

\overline{A^{'} B^{'}}

will no longer match with

\overline{P Q} .

Rotating A'B'C'D' about B' to map B'C' onto QR

Although the quadrilaterals have the same side lengths, no matter what rigid motion is applied,

A B C D

will not match

P Q R S .

The reason is that the quadrilaterals have different shapes.

\overline{A B} ≅ \overline{P Q} \overline{B C} ≅ \overline{Q R} \overline{C D} ≅ \overline{R S} \overline{D A} ≅ \overline{S P} \Rightarrow A B C D ≅ P Q R S

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.

Example

Determining Congruence Using a Sequence of Transformations

Consider the following pair of quadrilaterals $A B C D$ and $P Q R S$ in the coordinate plane.

Two quadrilaterals on the coordinate plane

Furthermore, consider the following sequences of transformations.

$S_{1} :$ Translation along $v = ⟨ 3, 1 ⟩$ followed by a $9 0^{\circ}$ clockwise rotation about the origin.
$S_{2} :$ Glide reflection defined by the line $y = - x$ and $v = ⟨ 3, - 3 ⟩ .$
$S_{3} :$ $18 0^{\circ}$ rotation about the origin followed by a reflection across $y = x .$

Are the given quadrilaterals congruent?

In the affirmative case, which of the listed sequences maps

A B C D

onto

P Q R S ?

Hint

Use the orientation of both quadrilaterals to discard one of the sequences. Then, apply the remaining sequences to $A B C D,$ one at a time, to find the image. Compare the image to $P Q R S .$

Solution

To determine if any of the given sequences maps $A B C D$ onto $P Q R S,$ apply them to $A B C D,$ one at a time, and compare each resulting image to $P Q R S .$

Sequence $1$

At first glance, it seems like the two quadrilaterals have opposite orientations. If this is true, the first sequence cannot map

A B C D

onto

P Q R S

because both translations and rotations maintain the orientation. To verify it, apply the sequence to

A B C D .

Sequence

1

maps only

\overline{A B}

onto

\overline{P Q} .

The remaining parts do not match. Consequently, Sequence

1

does not map

A B C D

onto

P Q R S .

Sequence $2$

A glide reflection is the composition of a reflection and a translation performed in any order. Since a reflection is involved, this sequence is a good candidate. To apply it to

A B C D,

start with the reflection.

Next, the translation along

v = ⟨ 3, - 3 ⟩

will be applied.

As can be seen, the given glide reflection mapped

A B C D

onto

P Q R S .

Therefore, the given quadrilaterals are congruent.

Sequence $3$

Like the previous sequence, this third sequence also involves a reflection and thus it is a good candidate. Here, the rotation is applied first. Since it is a

18 0^{\circ}

rotation, the direction of the rotation will not affect the resulting image.

Next, reflect the resulting image across the line

y = x .

As the diagram shows, this sequence does not map

A B C D

onto

P Q R S .

As a result, only the second sequence of transformations maps

A B C D

onto

P Q R S .

Pop Quiz

Practice Identifying Congruent Figures

In the following applet, the left-hand side polygon can be translated and rotated.

To translate the polygon, click on it and slide.
To rotate the polygon about $Q,$ click on any vertex of the polygon and rotate.

By applying these rigid motions, determine whether the given pair of polygons are congruent.

Illustration

Similar but Not Congruent

Because of copyright, in marketing, two different companies cannot have the same name or logo. Here, knowing how to determine if two figures are congruent is useful in dealing with intellectual property. Sometimes lesser-known brands use logos almost congruent to those of well-known brands. The intention is to receive instant recognition. For instance, consider the well-known logo of MathLeaks and the logo of a startup company called MathLovers.

Even though the two logos appear to be the same, there is no rigid motion that maps one logo onto the other. Therefore, they are not congruent. Consequently, there is no legal issue.

Closure

Congruence in Real Life

Congruent figures are everywhere in daily life. For example, the figures formed by lines on one half of a basketball court are congruent to the figures formed on the other half. However, verifying this congruence through rigid motions is a difficult thing to do, given the circumstance of not being able to see the rigid motions occur.

Therefore, it would be convenient to find a way of determining whether two figures are congruent without using rigid motions. The good news is that there is a simpler way for polygons, which will be studied in the next lesson.

Level 1

Level 2

Level 3

Rigid Motion and Congruence

The following figures are congruent.

Pair corresponding angles.

Pair corresponding sides.

Examining the figures, we see that they have both different directions and orientations. Therefore, to make them map onto each other, we must perform two rigid motions, a rotation, and a translation.

Rotation

Let's rotate XYZUV until it has the same orientation as ABCDE.

Translation

Finally, we will translate X'Y'Z'U'V' until it maps onto ABCDE.

Now we can pair corresponding angles. ∠ A&≅ ∠ X ∠ B&≅ ∠ V ∠ C&≅ ∠ U ∠ D&≅ ∠ Z ∠ E&≅ ∠ Y

In Part A, we were able to show how the figures would map onto each other. Let's take a look at that translation again.

We can see which pairs are corresponding sides. AB&≅ XV BC&≅ VU CD&≅ UZ DE&≅ ZY EA&≅ YX

For a pair of polygons, $A B C D ≅ F E H G .$ What rigid motion(s) maps $F E H G$ onto $A B C D ?$ Use as few transformations as possible.

two congruent quadrilaterals with different positions

two quadrilaterals which are congruent and have different orientations and positions

two quadrilaterals which are reflections of each other

Since the quadrilaterals look similar to squares, it might be difficult to identify corresponding vertices. However, we have been given a congruence statement. This helps us identify corresponding vertices by pairing letters with the same position within the given statement. A BC D ≅ F EH G [-1em] A→ F, B → E, C→ H, D→ G Let's highlight the corresponding vertices in the diagram.

We see that we neither need to perform a reflection nor a rotation to make the polygons map onto each other. We only need to make a translation.

Here, we see that the figures have different directions and orientations. We can try to perform both a rotation and a translation. Let's first perform a translation and make one pair of corresponding vertices map onto the other.

Finally, we can perform a rotation to make the figures map onto the other. Looks like it will work!

Looking at the diagram, we see that the quadrilaterals are mirror images of each other in a horizontal line of reflection. Therefore, we can map one shape to the other by reflecting it in a horizontal line that runs halfway between corresponding vertices.

Which of the polygons, $B$ through $G,$ are congruent to $A ?$

7 rectangles out of which three are congruent.

Examining the diagram, we see that all of the polygons are rectangles. To determine which figures are congruent to A, we will perform a series of rigid motions and check which map onto A. Let's first rotate all of the rectangles where the shorter side is not horizontal.

Next, we will perform a translation on A such that the bottom left corner matches the bottom left corner of the remaining rectangles. We will perform this translation one figure at a time.

As we can see, B and D are both congruent to A.

Which of the following statement(s) is/are true?

A : B : C : D : E : All rectangles are congruent . All circles are congruent . All squares are congruent . There are three main congruence transformations . Congruent shapes can be mapped onto each other through rigid motions .

Let's break down each statement.

Statement A and C

Two polygons are congruent if they have the same shape and size. Notice that all rectangles and squares are quadrilaterals with four right angles. However, this is not enough to determine congruence. For two quadrilaterals to be congruent, they must also have the same size. Rectangles and squares come in different sizes. Therefore, A and C are both false.

Statement B

Since there is only one dimension that decides the size of a circle, the radius, every circle is, in fact, similar to each other. However, this does not mean that two circles are congruent. The reason being two circles can have different radii.

Statement D

There are four different transformations a figure can undergo.

Translation
Rotation
Reflection
Dilation

Of these transformations, only the first three transformations are examples of rigid motions. These preserve shape and size which means they are what we call congruence transformations. Therefore, Statement D is true.

Statement E

Any pair of congruent shapes can be mapped onto the other by transforming one of the shapes until it maps onto the other shape. We can only perform this by using rigid motions, which preserve a shape's size and shape. Therefore, this is a true statement.