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7. Rigid Motion and Congruence
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Chapter 1
7. 

Rigid Motion and Congruence

Student Learning Objectives:
  • Identify and draw congruent figures
  • Differentiate between similar and congruent figures

Why
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.

Problem Solving Reasoning and Communication Error Analysis Modeling Using Tools Precision Pattern Recognition
Lesson Settings & Tools
11 Theory slides
6 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
Image Credits expand_more
Rigid Motion and Congruence
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.

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.

Applet to pick the correct 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 AB and any rigid motion. Let C be the image of A and D the image of B.

Segments AB and CD

Because rigid motions preserve distances, CD is equal to AB. Now, to check that every point of AB was actually mapped onto CD, consider a point P on AB different from the endpoints. Let Q be the image of P under the rigid motion.

Point P on AB and point Q not on CD

The idea is to show that Q lies on CD between C and D. To do this, it must be checked that CQ+QD is equal to CD. Again, since rigid motions preserve distances, CQ=AP and QD=PB. CQ=AP & (I) QD=PB & (II) Next, Equations (I) and (II) can be added together and simplified using the Segment Addition Postulate.

CQ=AP & (I) QD=PB & (II)
▼
Solve by elimination
SysEqnAdd

(I): Add II

CQ+ QD=AP+ PB QD=PB

(I): Segment Addition Postulate

CQ+QD=AB QD=PB
Substitute

(I): AB= CD

CQ+QD= CD ✓ QD=PB
Consequently, Q lies on CD which proves that the image of every point of AB lies on CD. 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.

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.
Image of the center of the circle C

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.

Image of the circle C
b As in Part A, start by finding the image of the center of C and the image of the given point.
Image of the center and image of the given point

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

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

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 A'B' and B'C'.

Perpendicular bisectors of A'B' and B'C'

The point of intersection of the perpendicular bisectors is the center of C'. That way, the image of C can be drawn.

Image of the three points of the circle C
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.

ABCDE ≅ JKLMN or CDEAB ≅ LMNJK

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.

Pair of possibly congruent figures

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.
Pointing Corresponding Vertices

Then, draw one of the figures on tracing paper and translate it so that the corresponding vertices match each other.

Translating one figure to match the corresponding vertices

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.

Rotating one figure to match corresponding sides

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.

Reflecting the figure along 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 AB onto PQ. However, the remaining parts of ABCD do not match the parts of PQRS. Equivalently, CD can be mapped onto RS 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 B'C' onto QR. However, by doing this, A'B' will no longer match with PQ.
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, ABCD will not match PQRS. The reason is that the quadrilaterals have different shapes. lAB ≅ PQ BC ≅ QR CD ≅ RS DA ≅ SP ⇒ ABCD ≅ PQRS 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.

A rectangle and and parallelogram with corresponding congruent sides
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 ABCD and PQRS in the coordinate plane.

Two quadrilaterals on the coordinate plane

Furthermore, consider the following sequences of transformations.

Are the given quadrilaterals congruent?

In the affirmative case, which of the listed sequences maps ABCD onto PQRS?

Hint
Use the orientation of both quadrilaterals to discard one of the sequences. Then, apply the remaining sequences to ABCD, one at a time, to find the image. Compare the image to PQRS.

Solution
To determine if any of the given sequences maps ABCD onto PQRS, apply them to ABCD, one at a time, and compare each resulting image to PQRS.

Sequence 1

At first glance, it seems like the two quadrilaterals have opposite orientations. If this is true, the first sequence cannot map ABCD onto PQRS because both translations and rotations maintain the orientation. To verify it, apply the sequence to ABCD.

Applying Sequence 1 to ABCD

Sequence 1 maps only AB onto PQ. The remaining parts do not match. Consequently, Sequence 1 does not map ABCD onto PQRS.

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 ABCD, start with the reflection.

Reflecting ABCD across y=-x

Next, the translation along v=⟨ 3,-3⟩ will be applied.

Translating the reflected image along vector v

As can be seen, the given glide reflection mapped ABCD onto PQRS. 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 180^(∘) rotation, the direction of the rotation will not affect the resulting image.

Rotating ABCD about the origin

Next, reflect the resulting image across the line y=x.

Reflecting the rotated image across y=x

As the diagram shows, this sequence does not map ABCD onto PQRS. As a result, only the second sequence of transformations maps ABCD onto PQRS.

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.

Random polygons
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.

Showing two almost equal logos
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.

Basketball Court
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
Exercise 3.1

The quadrilateral ABCD has undergone a sequence of rigid motions mapping it onto A'''B'''C'''D'''.

a preimage and an image that has undergone a series of rigid motions

The following are the sequence of transformations. Some key information has been omitted. Translation:& (x,y)→ (x+8,y) [0.15em] Rotation:& by counterclockwise [0.1em] & about the origin Reflection:& in the line Select the correct rotation and line of reflection.

Let's begin by performing the first transformation. It is a translation of 8 units to the right.

Examining the diagram, we can decipher that A'B'C'D' is a reflection of the final image A'''B'''C'''D''' in the x-axis. Recall the formula for reflecting something in the x-axis. Reflection in thex-axis (x,y) → (x,- y) The reflection in the x-axis changes the sign of the y-coordinate. We need to replicate a reflection in the x-axis using a combination of transformations, one rotation and one reflection. Let's have a look at some rules for rotating something counterclockwise about the origin.

Rotation Rule
90^(∘) (x,y) → (- y,x)
180^(∘) (x,y) → (- x,- y)
270^(∘) (x,y) → (y,- x)

We can see that a rotation of 180^(∘) changes the y-coordinate in the desired way. Therefore, let's perform this rotation.

Now we see that we need to perform a reflection in the y-axis, or x=0, to make A''B''C''D'' map onto A'''B'''C'''D'''.

Therefore, the transformations that are missing are a rotation of 180^(∘) counterclockwise about the origin and a reflection in x=0.

E
Hint & Answer
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Solution

Rigid Motion and Congruence

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