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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.

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.

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Answer

Hint

Solution