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1. Translations and Reflections
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Chapter 11
1. 

Translations and Reflections

Transformations like translations and reflections are essential for understanding the behavior of geometric objects in different contexts. These concepts reveal how figures can be moved or flipped while retaining their shape and size, making them crucial for analyzing symmetry, congruence, and spatial relationships. Translations help to explore the movement of objects in a plane, while reflections focus on understanding symmetry and mirroring across a line. These transformations have practical applications in design, navigation, and various problem-solving scenarios. Mastering these ideas supports the development of spatial reasoning and enhances comprehension of geometric principles, which are fundamental in mathematics and its real-world applications.
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15 Theory slides
15 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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Translations and Reflections
Slide of 15
There are lots of things that can be done to a given a geometric figure to obtain a new one. Since a figure is transformed into a new one, these are called transformations.
A quadrilateral ABCD and its image A'B'C'D' under a transformation

Transformations are useful to describe the relationship between two figures. This lesson will discuss two particular types of transformations: translations and reflections.

Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.

Explore

Moving a Triangle

In the following applet, arrow v dictates how △ ABC will move to get △ A'B'C'. Try changing the direction and length of the arrow to see different types of movements.

Notice how △ ABC and △ A'B'C' are congruent and how the orientation of △ A'B'C' matches the orientation of △ ABC. Furthermore, the segments AA', BB', and CC' are all parallel to arrow v, and all four segments have the same length.
Discussion

Transformations of Geometric Objects

A transformation is a function that changes a figure in a particular way — it can change the position, size, or orientation of a figure. The original figure is called the preimage and the figure produced is called the image of the transformation. A prime symbol is often added to the label of a transformed point to denote that it is an image.

A quadrilateral ABCD and its image A'B'C'D' under a transformation
Discussion

Translations of Geometric Objects

A translation is a transformation that moves every point of a figure the same distance in the same direction. To find the coordinates of a geometric object after a translation, a value a is added to the x-coordinate of every point of the preimage and a value b is added to the y-coordinate of every point of the preimage. (x,y) → (x+a,y+b) Positive a values indicate a translation to the right and positive b values correspond to a translation up.

Conversely, negative a values indicate a translation to the left and negative b values correspond to a translation down.

Translations preserve the side lengths and angle measures of the involved geometric objects.
Example

Placing a Pool in the Backyard

Zain's parents want to renovate their backyard. The lot is pretty wide and, once the weeds are taken care of, it could be used for plenty of things. The first thing that comes to Zain's mind is to put in a pool, so they make a sketch of the backyard to show their idea to their parents. They use the letter P to represent the pool.

Zain's dad says that the pool should not be right against the gate to the backyard, so he suggests two possible alternatives, P_1 and P_2.

a Let P be the preimage of the pool before a transformation. Which figure represents the image of P after a pure translation?

b The translation can be written as (x,y) → (x+a,y+b). Use the grid on the sketch to find the a and b values of the translation of the pool.

Hint
a A translation maps figures into congruent figures that have the same orientation.
b Focus on any corner of the pool. Count how many squares it is moved vertically and horizontally.

Solution
a Consider the pool location suggestions made by Zain's dad.

A pure translation takes a preimage P and makes it into an image that is congruent and has the same orientation as P. In the sketch, both P_1 and P_2 are congruent to P, so all that remains is to determine which image has the same orientation as P. Notice that the longest side of P is horizontal in the sketch.

This preimage will now be compared to images P_1 and P_2. The longest side of image P_1 is vertical in the sketch, so it does not have the same orientation as P. This means that P_1 is not a pure translation of P. On the other hand, P_2 matches P exactly.

Since P_2 is congruent to P and has the same orientation, it is a pure translation of P.

b Given the coordinates (x,y) of each point in a preimage, a translation of the point can be written in the following form.

(x,y) → (x+a,y+b) In this case, it is not necessary to know the coordinates (x,y), only the values a and b that are relevant to the translation. It was found in Part A that P_2 is the image of Zain's pool after a translation, so the focus will be in this image.

The a value of the translation corresponds to the horizontal displacement of the translation, while the b value is the vertical displacement. Since every point of the preimage is translated in the same way, any particular corner of the pool can be used along with the grid to find a and b. Just keep in mind that they have to be matching corners!

The horizontal displacement is 4 units to the right, so a=4. Likewise, the vertical displacement is 7 units up, so b=7.
Pop Quiz

Identifying Translations

In the following applet, △ ABC is translated to map onto △ A'B'C'. Write the a and b values of the translation.

Performing random translations to random triangles
Example

Replacing the Planter

The renovations are now taking place, so there is a lot going on in the backyard. During the morning, Zain's mom goes out to water some plants that were on a planter pot when she noticed that they are in the wrong place.

The planter was probably in the way when the builders were working on the pool. However, the plants need to be in the correct spot to get just the right amount of sunlight, so she kindly requests the workers to put them back where they belong when they are finished for the day. To find the original position of the pots, Zain placed a coordinate plane on the backyard plans.

Zain's mom asks the builders to translate the pot 7 units to the left and 1 unit up. Find the coordinates of the pot after the translation.

Hint
Find the initial coordinates of the pot. To translate it 7 units to the left and 1 unit up, subtract 7 from the x-coordinate and add 1 to the y-coordinate.

Solution
Zain's mom wants the builders to return the planter pot back to where it was, which they can do by translating it 7 units to the left and 1 unit up. Begin by finding the current coordinates of the pot.

The horizontal part of the translation is 7 units to the left. This means to subtract 7 from the current x-coordinate of the pot. 4 - 7=-3 On the other hand, the vertical part of the translation is 1 unit up, so add 1 to the current y-coordinate of the pot. -2 + 1 = -1 The x-coordinate of the pot after the translation is -3 and its y-coordinate is -1. (4-7,-2+1) = (-3,-1) The following applet can be used to visualize the translation.

Explore

Reflecting a Triangle

In the following applet, the vertices of △ ABC can be moved. The slider bar adjusts the slope of the line l. Once everything is set, reflect △ ABC across line l.

Is there any relationship between AA' and l? If so, do BB' and l have the same relationship? What about CC' and l?
Discussion

Reflections of Geometric Objects

A reflection is a transformation in which every point of a figure is reflected across a line. The line across the points are reflected in what is called the line of reflection. This acts like a mirror.

Triangle being reflected across a movable line
In more precise terms, a reflection across a line l maps every point A in the plane onto its image A' such that one of the following statements is satisfied.

  • If A is on the line l, then A and A' are the same point.
  • If A is not on the line l, then l is the perpendicular bisector of AA'.
Segment AA' intersects line ell perpendicularly, and line ell bisects segment AA'. Points B and B' coincide.
 Like translations, reflections also preserve side lengths and angle measures. However, reflections can change the orientation of the preimage.
Example

Reflections In the Pool Tiles

Once the pool was installed, Zain went to take a look at it. They noticed a unique pattern printed in the pool tiles.

Zain thinks of this as a transformation and writes the following labels for the vertices of both figures.

It seems that there is some sort of reflection between the tiles, but Zain is not completely sure. If there was a reflection, it would be possible to find a line of reflection. Help Zain find the line of reflection.

Answer

Hint
Draw segments that connect every point of the preimage with its corresponding point in the image — for example, CC'. The line of reflection is the perpendicular bisector of any segment connecting a point to its image.

Solution
When a point is reflected across a line, its image is such that the line of reflection is the perpendicular bisector of the segment that connects the point to its image. To find the line of reflection of the pool tiles, start by drawing a segment that connects a vertex and its image. For instance, draw CC'.

Next, construct the perpendicular bisector of CC'. This will represent the line of reflection used to make the pattern in the pool tiles.

With this, the line of reflection can be finally drawn.

Pop Quiz

Drawing Lines of Reflection

Drag the points in the following applet to draw the line of reflection used to map △ ABC onto △ A'B'C'. To do so, place the two points so they lie on the line of reflection. The measuring tool can be used to find the midpoint between corresponding vertices of the image and preimage, and they can also be used to find right angles.

Performing random reflections to random triangles

Extra

 How to Use the Measuring Tool
The measuring tool is useful for finding the midpoint between a point and its image. Place the ends of the measuring tool on a point and its corresponding image.

Next, move the middle point so that the angle of the measuring tool is 180^(∘) and the length of both segments is equal. This will make sure the middle point lies on the line of reflection.

Repeat this process on another pair of corresponding vertices to find a second point that lies on the line of reflection.

Now that two points that lie on the line of reflection have been identified, that line can be drawn.

Pop Quiz

Reflecting Triangles

Place points A', B', and C' where they should be to reflect △ ABC across the given line. The measuring tool can be used to find segments perpendicular to the line of reflection.

Performing random reflections to random triangles
Discussion

Reflections in the Coordinate Plane

In the coordinate plane, there is a particular relationship between the coordinates of a point and those of its image after a reflection across the coordinate axes. Investigate each relationship by using the following applet.

Applet to investigate the coordinates of a point after a reflection across the coordinate axes and the lines y=x and y=-x
Drawn from diagram, the following relations can be determined.

  • The image of (a,b) after a reflection across the x-axis is (a,- b).
  • The image of (a,b) after a reflection across the y-axis is (- a,b).
Example

Placing Chairs in the Backyard

It is always a good idea to have a picnic table close to the pool for snacks and drinks and to hang out while drying off, so Zain added this to the sketch, along with two chairs.

Zain noted that the table is placed along the x-axis they drew when trying to figure out where to return the planter pot.

To keep things symmetric, Zain wants to add two more chairs so that they are reflections across the x-axis of the other two chairs. Write the coordinates of the image of the left chair after a reflection across the x-axis.

Hint
Identify the coordinates of the left chair. A reflection across the x-axis can be plotted by changing the sign of the y-coordinate of the preimage.

Solution
Two more chairs are to be placed so that they are reflections across the x-axis of the currently shown chairs.

Start by finding the coordinates of the left chair.

To find the reflection of an object across the x-axis, just change the sign of its y-coordinate. (4, -1) → (4, 1) Now that the coordinates of the image have been identified, Zain can place the chairs!

The right chair can also be reflected by following the same procedure.

Closure

A Brief Talk About Glide Reflections

This lesson introduced two types of transformations of geometric objects. Translations move every point of the preimage the same distance in the same direction.

Reflections move every point in the preimage over to the line of reflection, which acts like a mirror.
These transformations can be combined to make a new transformation. When the figure is translated in a direction parallel to the line of reflection this is called a glide reflection. In the following applet, △ ABC is translated in the direction and length of arrow v, and then is reflected across the line l.
Performing a Glide Reflection on a Triangle
A glide reflection can also be done by performing the reflection before the translation.
Performing a Glide Reflection on a Triangle

Note that the image is the same in both cases, which means that, in the case of a glide reflection, the image does not depend on the order of the transformations.

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Translations and Reflections
Exercise 3.1

A point P(x,y) is translated twice to obtain the point P''. The first translation is 2 units left and 1 unit down, and the second translation is 2 units right and 1 unit up. Find the coordinates of P'' without graphing.

We are told that a point P(x,y) is translated twice. The first translation is 2 units left and 1 unit down. Let's write this using translation notation. P(x,y) → P'(x-2,y-1) After this translation, we get a new point whose coordinates are (x-2,y-1). This point is translated again, this time 2 units right and 1 unit up. We can write this translation using translation notation as well, but keep in mind that the point being translated is P'(x-2,y-1), not the original point P(x,y). P'(x-2,y-1) → P''((x-2)+2,(y-1)+1) Looking at the x-coordinates of both translations at once, we can see that this is essentially subtracting 2 from x and then adding 2 to the result. This gives us back x. x-2+2 = x Similarly, we subtract 1 from y and then add 1 to the result. y-1+1 = y This means that after both translations, we are left back with the original point. The coordinates of the twice-translated point P'' are (x,y).

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Hint & Answer
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Solution

Consider the given triangle.

Triangle ABC is translated 3 units to the left, and then reflected over the x-axis. Identify the coordinates of the vertices of the triangle after the two transformations.

Let's take a look at the given triangle.

We are asked to find the coordinates of the vertices of the triangle after translating it 3 units left and then reflecting it over the x-axis. Let's begin by labeling the coordinates of the vertices A, B, and C.

We want to translate this triangle 3 units to the left. This means that we should subtract 3 units from the x-coordinate of each of its vertices.

Original Coordinates Translation Translated Coordinates
A(1,1) (1-3,1) A'(-2,1)
B(2,3) (2-3,3) B'(-1,3)
C(3,1) (3-3,1) C'(0,1)

Let's see the image of △ ABC after the translation.

Next, we want to reflect this image across the x-axis. We can do this by simply changing the sign of the y-coordinate of each vertex.

Translated Coordinates Reflected Coordinates
A'(-2,1) A''(-2,-1)
B'(-1,3) B''(-1,-3)
C'(0,1) C''(0,-1)

Finally, we can sketch the image of △ ABC after both transformations. Note that we use two primes because there were two transformations involved.

The coordinates of the image after the reflection are A''(-2,-1), B''(-1,-3), and C''(0,-1).

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Hint & Answer
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Solution

A point (x,y) is reflected over the x-axis, then reflected over the y-axis. What are the coordinates of the point after both reflections?

We are told that a point (x,y) is reflected over the x-axis, then again over the y-axis. Let's consider the first reflection. We know that when we reflect a point over the x-axis, the x-coordinate stays the same and the sign of the y-coordinate is changed. (x,y) → (x, - y) This means that the image of the point (x,y) after the first reflection is (x,- y). We now need to reflect this image across the y-axis. This time the y-coordinate stays the same but the sign of the x-coordinate changes. (x, - y) → (- x, - y) This means that the image of the point (x,y) after both reflections is (- x, - y).

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Hint & Answer
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Solution

Translations and Reflections

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