Exploring Transformations: Translations and Reflections in Geometry

In the following picture, the preimage $P$ is shown in the center along with several images $P_{1},$ $P_{2},$ and so on.

Which of the images corresponds to a pure translation of the preimage

P

A pure translation moves the preimage into an image that has the same orientation as the preimage. Let's now take a look at the given picture.

Notice that P looks like an arrow pointing down. Since a pure translation preserves the orientation of the preimage, we should look for an image that also looks like an arrow pointing down.

Image P_8 has the same size, shape, and orientation as P. This means that P_8 is the image of P after a pure translation.

The flight of a bird was simulated by an artist using stop-motion animation. This illustration describes the flight of the bird from point $A$ to point $B$ and then from point $B$ to point $C .$

The translation can be written as

(x, y) \to (x + a, y + b) .

Find the

a

value of the translation from point

A

to point

B .

Write the

b

value of the translation from point

B

to point

C .

We are given three positions in the flight of the bird — A, B, and C. We want to describe the translation from point A to point B using translation notation. Let's take a look at the given picture and focus on the coordinates of points A and B.

In a translation, the x-coordinate of the preimage changes by the value of the horizontal translation a. The y-coordinate of the preimage changes by the value of the vertical translation b. (x,y) → (x+ a,y+ b) Since we are interested in the movement from point A to point B, the preimage is A and the image is B. The preimage is at point (- 2, - 2), so let's substitute - 2 for x and - 2 for y. ( - 2, - 2) → ( - 2+a, - 2+b) The image is at point (1, -1). This means that (-2 + a, -2 + b) = (1,-1). Let's write an equation with the x-coordinates and solve for a.

This means that the a value of the translation is 3.

This time we want to describe the translation from point B to point C.

This time B is the preimage. ( 1, -1) → ( 1+a, -1+b) The image C is at point (2,2), so we can write an equation like we did in Part A. (1+a,-1+b) = (2,2) Since we are interested in the b value of the translation, we will write an equation with the y-coordinates and solve it.

The b value of the translation is 3.

Zain went out for a walk. They started from their house at point $A$ and went all the way to the park, represented by point $A^{'} .$

This translation can be written as

(x, y) \to (x + a, y + b) .

Find the

b

value of the translation from point

A

to point

A^{'} .

We are asked to find the b value that describes the translation from point A to A'. Let's take a look at the given diagram.

Notice that this translation is purely horizontal. In a purely horizontal translation, the x-coordinate of the preimage changes by the value of the horizontal translation a, but the y-coordinate does not change. In other words, the y-value remains the same. (x,y) → (x+a,y) There is no need to write an equation for the y-coordinate since it does not change. The b value of this translation is 0. b=0

Consider $△ A B C$ with vertices $A (- 3, 3),$ $B (- 2, 2),$ and $C (1, 2) .$

Let

△ A^{'} B^{'} C^{'}

be the image of

△ A B C

after a translation

2

units right and

4

units down. Identify the coordinates of

A^{'},

B^{'},

and

C^{'} .

Let's take a look at the given triangle.

Translations are performed by adding or subtracting values from the x-coordinate if the figure is being moved left or right, and from the y-coordinate if the figure is being moved up or down. To find the coordinates of the image of our figure after a translation 2 units right and 4 units down, we will add 2 to the x-coordinate and subtract 4 from the y-coordinate of each vertex.

Vertices of ABC	(x+2,y-4)	Vertices of A'B'C'
A(- 3,3)	(- 3 + 2,3 - 4)	A'(-1, -1)
B(-2,2)	(- 2 + 2,2 - 4)	B'(0,-2)
C(1,2)	(1 + 2,2 - 4)	C'(3, -2)

We found the vertices of the transformed triangle! We can also visualize the translation in the following applet.

Consider the quadrilateral $A B C D$ with vertices $A (5, 3),$ $B (3, 1),$ $C (1, 3),$ and $D (3, 5) .$

Let

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

be the image of

A B C D

after a translation

4

units left and

3

units down. Identify the coordinates of

A^{'},

B^{'},

C^{'},

and

D^{'} .

Let's take a look at the given quadrilateral.

Translations are done by adding or subtracting values from the x-coordinate if the figure is being moved left or right, and from the y-coordinate if the figure is being moved up or down. We want to find the image of our figure after a translation 4 units left and 3 units down. Let's subtract 4 from the x-coordinate and subtract 3 from the y-coordinate of each vertex.

Vertices of ABCD	(x-4,y-3)	Vertices of A'B'C'D'
A(5,3)	(5 - 4,3 - 3)	A'(1, 0)
B(3,1)	(3 - 4,1 - 3)	B'(-1,-2)
C(1,3)	(1 - 4,3 - 3)	C'(-3, 0)
D(3,5)	(3 - 4,5 - 3)	D'(-1,2)

We can visualize the translation in the following applet.

The following picture contains a hexagon along with its reflection across a line of reflection.

Which of the following graphs shows the line of reflection?

When a point is reflected across a line, its image is located such that the line of reflection is the perpendicular bisector of the segment that connects the point to its image.

Since we are reflecting a figure across a line of reflection, the above reasoning must apply to every point of the figure. This means that we have to find the parallel segments that connect the vertices of the hexagons. Here, we do not need to know which hexagon is the preimage and which one is the image — we can just draw the segments.

Now we choose one of the segments we drew before moving to the next step. We can use any of the segments, but this solution will use the bottom right vertex of the top hexagon.

We will now draw the perpendicular bisector of this segment.

This perpendicular line is the line of reflection.

Let's look for the graph that has this same line.

This means that the answer is A.

Zain once read in a book that rainbows are actually circular and that we just see the regular rainbow shape because we are not seeing them from a vantage point that is high enough above the ground.

To complete the circle, it would be necessary to perform a reflection across the

x -

axis of what is shown in the picture. Identify the coordinates of the reflected points

A^{'},

B^{'},

and

C^{'} .

Consider the given sketch of a rainbow.

We want to reflect points A, B, and C on this rainbow over the x-axis. Since the reflection is over the x-axis, we will simply change the sign of the y-coordinate of each point. (x,y) → (x,- y) Let's find the reflections of the given points!

Points on the Preimage	Points on the Image
A(-3.5,0)	A'(-3.5,0)
B(-2.5,2.5)	B'(-2.5,-2.5)
C(0,3.5)	C'(0,-3.5)

Let's see how the reflected points look in the coordinate plane.

Notice that A and A' are the same point. This is because A lies on the line of reflection. We will now finish the whole rainbow according to what Zain read in the book. Imagine that the x-axis is a mirror and use it to draw the rest of each colored circle.

This is how a rainbow would look if we were to see one from high enough above the ground!

For each given pair of points, determine whether the image corresponds to a reflection across the $x -$ axis, the $y -$ axis, or neither.

A (3, 5) and A^{'} (3, - 5)

B (1, - 2) and B^{'} (- 2, - 2)

C (- 1, - 1) and C^{'} (1, - 1)

We want to determine whether the given point A'(3,- 5) is a reflection of A(3,5) across an axis. Let's start by drawing both points on the coordinate plane.

Next, we draw a segment connecting both points.

Notice that this segment is perpendicular to the x-axis, so we might assume that A' is as a reflection over the x-axis. This assumption is correct only if the axis also bisects the segment we drew.

Since the distances between each point and the axis are the same, we know that the x-axis is the perpendicular bisector of the segment that connects A and A'. This means that A' is the reflection of A over the x-axis.

Now let's plot B(1,-2) and B'(-2,-2) and the segment that connects them.

This time the segment is perpendicular to the y-axis, but notice that the axis does not bisect our segment.

This means that this transformation is not a reflection across the y-axis, so the answer is neither.

Finally, let's plot C(-1,-1) and C'(1,-1) along with a segment connecting them.

The y-axis is the perpendicular bisector of the segment we drew.

This means that C' is the reflection of C over the y-axis.

Translations and Reflections

Catch-Up and Review

Moving a Triangle

Transformations of Geometric Objects

Translations of Geometric Objects

Placing a Pool in the Backyard

Hint

Solution

Identifying Translations

Replacing the Planter

Hint

Solution

Reflecting a Triangle

Reflections of Geometric Objects

Reflections In the Pool Tiles

Answer

Hint

Solution

Drawing Lines of Reflection

Extra

Reflecting Triangles

Reflections in the Coordinate Plane

Placing Chairs in the Backyard

Hint

Solution

A Brief Talk About Glide Reflections

Translations and Reflections

Recommended exercises

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