TRANSFORMATIONS AND CONGRUENCE
INTRODUCTION
The objective for this lesson on transformations and congruence is, the student will prove congruence by identifying the sequence of transformations of figures in order to solve real world problems.
The skills students should have in order to help them in this lesson include translations, reflections and rotations.
We will have three essential questions that will be guiding our lesson. Number one, what types of transformations result in congruent figures? Explain your thinking. Number two, how is the ability to identify transformations helpful in proving congruence of figures? And number three, why is it valuable to recognize congruence through rigid transformations? Justify your thinking.
We will begin by completing the warm-up identifying rigid transformations to prepare for transformations and congruence in this lesson.
SOLVE PROBLEM – INTRODUCTION
The SOLVE problem for this lesson is, Barbara has been adjusting the placement of some geometric wall decals. She is using a coordinate grid on the wall to assist. She began with Figure A and by the time she completed some transformations, the decal ended in the position represented by Figure A prime. What sequence or sequences of rigid transformations are possible for the adjustment of the triangular decal?
A picture of the coordinate grid provided in this question is seen here.
We will begin by Studying the Problem. First we need to identify where the question is located within the problem and we will underline the question. The question for this problem is, what sequence or sequences of rigid transformations are possible for the adjustment of the triangular decal?
Now that we have identified the question we want to put this question in our own words in the form of a statement. This problem is asking me to find the sequence of rigid transformations possible for the adjustment of the triangular decal.
During this lesson we will learn how to prove congruence through identifying sequence of rigid transformations to solve problems. We will use this knowledge to complete this SOLVE problem at the end of the lesson.
REVIEW OF RIGID TRANSFORMATIONS
Let’s start by reviewing what we know about Rigid Transformations.
The rigid transformation of sliding a figure is a translation. The rigid transformation of turning a figure is a rotation. And the rigid transformation of flipping a figure over an axis is a reflection.
When a rigid transformation occurs, the shape and size of the figure remain the same. The original figure and the transformed figure are congruent figures.
In the previous lesson, we learned from measuring line segments and angles of figures that rigid transformations do not change size and shape. Therefore, congruency is maintained when any one of these transformations occur.
Therefore, if we can show that the change from the pre-image to the image is one of the three previously discussed rigid transformations, then we know that the figures will be congruent because by definition they would not have changed shape or size when reflected, rotated or translated.
We will use this knowledge to prove congruency with different transformations within the lesson.
CONGRUENCY WITH TRANSLATIONS
Take a look at the transformation shown on the coordinate grid. What transformation appears to have occurred from Triangle ABC to Triangle A Prime, B Prime,C Prime? It appears to be a translation. The figure appears to have slid from one place on the coordinate grid to another.
Do the measures of the line segments change with a translation? We know from the previous lesson that no, the line segments do not change length with a translation.
Do the measures of the angles change with a translation? We know from the previous lesson that no, the angles do not change size with a translation.
Therefore, if we can prove that a translation occurred, we know that the figures are congruent.
Now, let’s trace Triangle ABC on a sticky note and label the vertices on the inside of the triangle. Now, cut the triangle and use the triangle for exploring during the activity!
You may not need your triangle for this first activity, but keep it close by for other activities.
Now, take a look at the transformation shown on the coordinate grid. What are the coordinates of point A? They are negative six, five.
What are the coordinates of point B? They are negative six, one.
And what are the coordinates of point C? They are negative two, one.
Let’s include the coordinates for Triangle ABC in the graphic organizer. For points A, B, and C.
Next let’s look at A Prime, B Prime, C Prime.
What are the coordinates of point A Prime? One, six
What are the coordinates of point B Prime? One, two
And what are the coordinates of point C Prime? Five, two
Let’s include the coordinates for triangle A Prime, B Prime C Prime in the graphic organizer as well. For points A Prime, B Prime and C Prime.
Now let’s talk about the movement of triangle ABC to triangle A Prime, B Prime C Prime.
What is the vertical change from Point A to Point A Prime? The vertical change was up one unit.
What was the vertical change from Point B to Point B Prime? It was also moved up one unit.
And what is the vertical change from Point C to Point C Prime? Again, the point moved up one unit.
So what do you know about the translation vertically for the vertices? Each vertex moved up one unit.
Notice that you can easily count the units between points, but you can also subtract the difference in the x-values and the y-values to find the change horizontally and vertically for this step and the next step.
Now let’s take a look at the horizontal change in the Points from triangle ABC to triangle A Prime, B Prime, C Prime.
What is the horizontal change from Point A to Point A Prime? The point moved right seven units.
What is the horizontal change from Point B to Point B Prime? It also moved to the right seven units.
And what is the horizontal change from Point C to Point C Prime? Again, the point moved to the right seven units.
So what do you notice about the translation horizontally for the vertices? Each vertex moved right seven units.
What can you conclude about Triangle ABC and Triangle A Prime, B Prime, C Prime? Knowing that a translation occurred because all of the vertices translated vertically and horizontally the same number of units, we know that both triangles are congruent.
CONGRUENCY WITH REFLECTIONS
Now take a look at the transformation shown here on the coordinate grid. What transformation appears to have occurred from Triangle ABC to Triangle A Prime, B Prime, C Prime? It appears to be a reflection over the x-axis.
Do the measures of the line segments change with a reflection? We know from the previous lesson that no, the measures of the line segments do not change with a reflection.
Do the measures of the angles change with a reflection? We know from the previous lesson that no the measures of the angles do not change with a reflection.
Therefore, if we can prove that a reflection occurred, we know that the figures are congruent.
Let’s look at triangle ABC. What are the coordinates of point A? The coordinates are negative six, five. What are the coordinates of point B? The coordinates of point B are negative six, one. And what are the coordinates of point C? Negative two, one.
Let’s record this information in the graphic organizer for triangle ABC.
Now let’s look at triangle A Prime, B Prime, C Prime. What are the coordinates of point A Prime? Negative six, negative five. What are the coordinates of point B Prime? Negative six, negative one. And what are the coordinates of point C Prime? Negative two, negative one.
Let’s also include this information in the graphic organizer for triangle A Prime, B Prime, C Prime.
Now looking at the coordinates of triangle ABC and triangle A Prime, B Prime, C Prime, what do you notice about the x-coordinates of the figures? The x-coordinates of the pre-image are the same as the image.
What do you notice about the y-coordinates of the figures? The y-coordinates of the image are the opposite of the y-coordinates of the pre-image.
So what can you conclude about the Triangle ABC and Triangle A Prime, B Prime, C Prime based on the coordinates? We know that a reflection over the x-axis has occurred because all of the x-coordinates remained the same while the y-coordinates have opposite signs. With a reflection occurring, we know that both triangles are congruent.
CONGRUENCY WITH ROTATIONS
Take a look at the transformation shown here on the coordinate grid. What transformation appears to have occurred from Triangle ABC to Triangle A Prime, B Prime, C Prime? It appears to be a one hundred eighty degree rotation about the origin.
Do the measures of the line segments change with a rotation? No, we know from the previous lesson that the measures of the line segments stay the same with a rotation.
Do the measures of the angles change with a rotation? No, we also know from the previous lesson that the measures of the angles stay the same with a rotation.
Therefore, if we can prove that a rotation occurred, we know that the figures are congruent.
How do we rotate a figure about the origin in the previous lesson? We would trace a figure on a sticky note and hold the corner of the note down at the origin, completing the rotation.
Instead of using the sticky note, we can simply rotate the page of the book one hundred eighty degrees. If we rotate the book one hundred eighty degrees we should have a figure that matches the exact coordinates of the transformed figure.
Let’s do that now. We will rotate the page of the book one hundred eighty degrees. What do you notice happened? The original triangle and the transformed triangle have switched places.
Triangle ABC is now located in quadrant four, when we rotate the page of the book one hundred eighty degrees.
Let’s look at triangle ABC, which is now locates in quadrant four. What are the coordinates of point A? They are six negative five. What are the coordinates of point B? They are six, negative one. And what are the coordinates of point C? They are two, negative one. Let’s record this information in the graphic organizer under the column on the left hand side labeled Rotated Triangle ABC.
Now let’s rotate the book back to its original position seen here. Now we are going to look at triangle A Prime, B Prime, C Prime, which is located in quadrant four.
What are the coordinates of point A Prime? Six, negative five. What are the coordinates of point B Prime? Six, negative one. And what are the coordinates of point C Prime? Two, negative one. Let’s record this information in our graphic organizer in the right hand column for Triangle A Prime, B Prime, C Prime.
Now let’s talk about what we notice. What do you notice about the x-coordinates of the figures in the graphic organizer? The x-coordinates after the rotation match the x-coordinates of Triangle A Prime, B Prime C Prime.
What do you notice about the y-coordinates of the figures? The y-coordinates after the rotation match the y-coordinates of Triangle A Prime, B Prime, C Prime.
So what can you conclude about Triangle ABC and Triangle A Prime, B Prime, C Prime based on the coordinates? We know that a rotation about the origin occurred because all of the coordinates of the turned Triangle ABC match all of the coordinates of Triangle A Prime, B Prime, C Prime. With a rotation about the origin occurring, we know that both triangles are congruent.
We are now going to take a look at another transformation for Triangle ABC. It is seen here on the coordinate grid. What transformation appears to have occurred from Triangle ABC to Triangle A Prime, B Prime, C Prime? It appears to be a one hundred eighty degree rotation about Point C.
Do the measures of the line segments change with a rotation? No.
Do the measures of the angles change with a rotation? No.
Therefore if we can prove that a rotation about Point C occurred, we know that the figures are congruent.
How did we rotate a figure about a point in the previous lesson? We would trace the figure on a sticky note and cut out the shape only. Then we would hold the shape down at the point of rotation, completing the rotation.
Take out the traced triangle and compete the rotation. Start by placing your traced triangle on top of triangle ABC and then complete the rotation about point C, to rotation the figure one hundred eighty degrees.
What do you notice happened? The original triangle and the transformed triangle have switched places.
So what are the coordinates of point A after the rotation? After the rotation the coordinates of point A on our sticky note are two, negative three. What are the coordinates of point B after the rotation? The coordinates of point B after the rotation, looking at our sticky note, are two, one. And what are the coordinates of point C after the rotation? The coordinates of point C after the rotation using our sticky note are negative two, one.
Let’s record this information in the graphic organizer under the first column labeled Rotated Triangle ABC.
Now let’s take a look at triangle A Prime, B Prime, C Prime on the original transformation on the coordinate plane. We will not need our sticky note for this part of the activity.
What are the coordinates of Point A Prime? Two, negative three. What are the coordinates of Point B Prime? Two, one. And what are the coordinates of Point C Prime? Negative two, one.
Let’s record this information in the right hand column of our graphic organizer labeled Triangle A Prime, B Prime, C Prime.
Now let’s talk about the information that we’ve gathered in the graphic organizer regarding the coordinates of Triangle ABC and Triangle A Prime, B Prime, C Prime.
What do you notice about the x-coordinates of the figures? The x-coordinates after the rotation match the x-coordinates of Triangle A Prime, B Prime, C Prime.
And what do you notice about the y-coordinates of the figures? The y-coordinates after the rotation match the y-coordinates of Triangle A Prime, B Prime, C Prime.
So what can you conclude about Triangle ABC and Triangle A Prime, B Prime, C Prime based on the coordinates? We know that a rotation about Point C occurred because all of the coordinates of the turned Triangle ABC match all of the coordinates of Triangle A Prime, B Prime, C Prime. With a rotation about Point C occurring, we know that both triangles are congruent.
IDENTIFYING SEQUENCES OF TRANSOFRMATIONS
The transformation seen here will be used to represent Route one in the following activity. Look at Route one in the left column, take a moment to discuss these figures. What transformation do you think is happening to get from the pre-image to the image? When comparing the pre-image to the image we can compare basic characteristics of transformations. Just because a transformation doesn’t appear to be the correct one immediately, doesn’t mean it won’t be a useful transformation later.
Looking at the images, do they appear to be a mirror image of each other? No. What does this mean? It means that for right now, we can eliminate a reflection.
Can we simply slide Triangle ABC to match the other figure and have them line up exactly? No. So what does this mean? It means for right now, we can eliminate a translation.
What transformation is left? A rotation. If we want to get the triangle from the second quadrant to the fourth quadrant, as we see on the coordinate grid here, how should we rotate it? We should rotate it one hundred eighty degrees about the origin.
When we rotate triangle ABC one hundred eighty degrees about the origin, our rotated triangle would be here, where you see the green triangle now on the coordinate grid. We will represent this triangle as triangle A Prime, B Prime, C Prime.
Now looking at Triangle A Prime, B Prime C Prime and Triangle A Double Prime, B Double Prime, C Double Prime, are they mirror images? No. So what does this mean? It means that we can eliminate a reflection for right now.
Can we simply slide Triangle A Prime, B Prime, C Prime to match the other figure and have them line up exactly? Yes we can.
So how should we slide the figure, A Prime, B Prime, C Prime to match up with figure A Double Prime, B Double Prime, C Double Prime? We can translate the triangle A Prime, B Prime, C Prime one unit. We can translate triangle A Prime, B Prime, C Prime up one unit and to the left two units.
If you complete this translation, does Triangle A Prime, B Prime, C Prime end with Triangle A Double Prime, B Double Prime, C Double Prime? Yes.