Translations Honors Geometry A Unit 5: Rigid Transformations

The pre-image of a triangle has vertices at (-2,3), (-4,-1), and (-1,-1). If this triangle is translated such that the vertex at (-2,3) is translated to the point (1,4), what are the coordinates of the other two vertices?

(-4,-1) goes to(-1,0) and (-1,-1) goes to (2,0).

A triangle that has vertices at (4,2),(2,0),and(5,1)(4,2),(2,0),and(5,1) is rotated 270°270° counter-clockwise about the origin. What are the coordinates of the vertices of the image?

(4,2) goes to (2,−4),(2,0) goes to (0,−2),and(5,1) goes to (1,−5)

What mapping represents a counterclockwise rotation of the pre-image (x,y) about the origin by 270°2?

(x,y)→(y,−x)

What mapping represents a clockwise rotation of the pre-image (x,y) about the origin by 180°?

(x,y)→(−x,−y)

What degree of rotation will map an equilateral triangle to itself?

120

How many lines of symmetry does a nonsquare rhombus have?

2

center of rotation

A fixed point around which shapes move in a circular motion to a new position.

vector

A quantity that has both size and direction.

Performing a reflection about y=x, and then reflecting the result about y=−x, or vice versa, is the same as performing which of the following single rigid motions?

A rotation by 180°

Performing a reflection about y=x, and then reflecting the result about y=−x, or vice versa, is the same as performing which single rigid motion?

A rotation by 180°.

How many lines of symmetry does a square have?

A square has four lines of symmetry; a horizontal line through the center, a vertical line through the center, and the two diagonals.

True or false? To draw the image of a figure after a translation, reflection, or rotation, we have to apply the transformation to every point in the figure.

False. We only need to apply the transformation to the vertices of polygons and the centers of circles. Then, we can redraw circles and line segments to construct the full image. The reason we can do this is that the transformations preserve shape and size, so that the pre-image vertices and centers map to the vertices and centers of the image.

Explain how learning multiple methods could help you adapt to change in later life.

Imagine you are working at a job where you are an expert in one way to do a particular task. Your boss then pairs you up with a new coworker who has a different way to do the same task. That coworker, sadly, refuses to learn your method. Unless you adapt to change, by learning their method, even though you have your own, you may run into difficulties working with this person. However, if you take the initiative to adapt to this change, by learning a new method for the same task, you can learn their method and be better able to work with the new person. In addition, you may find benefits in this other method, and when you know both methods, you can better decide the right one for the situation.

What does it mean for a transformation to carry a figure onto itself?

It means the image lines up perfectly with the pre-image.

Where does the negative y-axis rotate to for a 270°counter-clockwise rotation?

It rotates onto the negative x-axis.

You can perform a translation, reflection, or rotation on a circle using the same mappings, by applying the mapping to the center of the circle and drawing a new circle at the new center that has the same radius. Why does this work?

Just like you can draw the image of a translation, reflection, or rotation of a polygon by applying a mapping to each vertex, you can draw the image of a translation, reflection, or rotation of a circle by applying a mapping to its center. This is because a translation, reflection, and rotation preserves size and shape, so that the image will be a circle with the same radius, but with a center that has been moved according to the transformation.

rigid motions

Movement (up, down, left, right, diagonal, flip, etc.) without adjusting the shape of a figure.

What is a sequence of transformations?

Multiple transformations applied one after the other.

Rosalita translates a triangle in the coordinate plane and obtains a triangle with different angle measures. Is this possible?

No, angle measures are preserved during a translation, so Rosalita must have made a mistake.

Carlton says that tipping his drinking glass on its side is an example of a translation because the base of the glass is in a different location. Is he correct?

No, he is not correct. If he slid the glass across the table, it would be an example of a translation. But because the glass is now in a different orientation, his transformation is not an example of a translation.

Is it possible to list all rigid motions that produce the same mapping? Why or why not?

No, it is not possible. Consider that you can rotate by 90°, then rotate by 270°, to get a net effect of (x, y)→(x, y) Therefore, you could add the sequence "rotate by 90°, and then rotate by 270°" as many times as you want without changing the net result. So, it would be impossible to list every possible sequence of rigid motions that produces the same mapping.

Two rays form a 53° angle. If this angle is translated 4 units up and to the right, does the measure of the angle change?

No, the measure of the angle does not change. Translations do not affect the measure of an angle.

Is there only one rigid motion that will carry a figure onto another figure that is congruent to it?

No. For any rigid motion, there are infinitely many sequences of rigid motions that will have the same end result. But in practice, we only need to find one.

image

Post translation of an image

Write a series of two transformations, without using rotations, that produce a rotation of 180°, and use mapping notation to prove it.

Reflecting about the y-axis makes the x-coordinate negative. Reflecting the result about the x-axis will make the y-coordinate negative, which results in the same mapping as rotating by 180°.

Why can rigid motions be used to determine congruence?

Rigid motions preserve side lengths and angle measures. Any number of rigid motions produces a congruent image. So if you can establish that there is a sequence of rigid motions that takes one figure onto another, you have established that they are congruent. Likewise, if you can establish that no rigid motion takes one figure onto another, then the figures are not congruent.

Write the following sequence of rigid motions as a single mapping on a point (x, y). Rotate by 90°, translate by (−1, −1), rotate by 90°.

Rotate by 90°: (x, y)→(−y, x) Translate by (−1, −1): (−y, x)→(−y−1, x−1) Rotate by 90° : (−y−1, x−1)→(−(x−1), −y−1)=(−x+1, −y−1) Therefore, the effect of the entire sequence is (−x+1, −y−1), which is the same as rotating by 180°, and then translating by (1, −1)

How can rigid motions be used to determine congruence?

Since the image of a rigid motion is congruent to the pre-image, if you can establish a sequence of rigid motions that produces a figure from another figure, then you have established the congruence of the two figures.

Explain the term rigid motion.

Something that is rigid cannot be stretched, bent, or deformed. Motion describes a change in location. A rigid motion is a transformation that moves, or reorients a pre-image, without deforming its shape or changing its size. The pre-image is kept rigid as it is transformed into the image.

Write 5 degrees, which if rotated by, would carry any figure onto itself.

The first 5 rotations are 360°, 720°, 1080°, 1440°, and 1800°.

What are the coordinates of the points (−2,1) and (-3,4) after a 180° rotation?

The image of (-2,1) is (2,-1), and the image of (−3,4) is (3,−4).

The image of a rigid motion is ____ to the pre-image.

The image of a rigid motion is congruent to the pre-image.

The pre-image of a translation is a line. How is the image line related to the pre-image line?

The lines are parallel.

What is the mapping for a reflection of a point (x, y) about the line y=x?

The map is (x, y)→(y, x)

What is the mapping for a reflection of a point (x, y) about the line y=−x?

The map is (x, y)→(−y, −x).

Draw a triangle with vertices A(2,2), B(5,2), and C(5,4). Then, rotate the triangle 180°. Show your work using mapping notation.

The mapping that represents a rotation by 180° is (x,y)→(−x,−y) where both coordinates are negated.

Draw a triangle with vertices A(2,2), B(5,2), and C(5,4) Then, reflect the triangle about the x-axis. Show your work using mapping notation.

The mapping that represents reflection about the x-axis is (x,y)→(x,−y) where the y-coordinate is negated. A(2,2)→(2,−2)=A`B(5,2)→(5,−2)=B`C(5,4)→(5,−4)=C`

Rotate the triangle with vertices A(2,2), B(5,2), and C(5,4) 270°. Graph the pre-image and image with the vertices labeled with their coordinates.

The mapping that represents rotating by 270° is (x,y)→(y,−x), where the x-coordinate is negated, and the two coordinates are swapped.

Besides the distance of the image from the pre-image, what else can help you determine which sequence of transformations produces a given image from a given pre-image?

The orientation of the image helps to determine which transformations must have been applied. Knowing how reflections and rotations change the orientation of an image will be very helpful for deducing which transformations to apply in the sequence.

David wants to rotate the point (1,1)(1,1) 90°90° counter-clockwise about the point (2,2)(2,2). He says that for a 90°90°counter-clockwise rotation, the point (x,y)(x,y) becomes (−y,x)(-y,x). So the image of the point is (−1,1)(-1,1). What error did David make?

The rule that David used applies only to rotations about the origin.

preserves

areas that are protected by various means to maintain biodiversity

transformation

process in which one strain of bacteria is changed by a gene or genes from another strain of bacteria

clockwise

the circular direction in which hands in an analog clock move

counter-clockwise

the circular direction opposite to clockwise

pre-image

the original figure in a transformation

True or false? Rigid motions preserve the orientation of the pre-image.

False, only translations preserve the orientation of the pre-image.

Rotate the triangle with vertices A(2,1)A2,1, B(5,1)B5,1, and (5,3)5,3 270°270° counterclockwise. Then, graph the pre-image and image with the vertices labeled with their coordinates.

A=2,1→1,-2=A'B=5,1→1,-5=B'C=5,3→3,-5=C'

How many lines of symmetry does an equilateral triangle have?

An equilateral triangle has 3 lines of symmetry; each altitude of the triangle acts as a line of symmetry.

Why can rigid motions establish congruence?

Because rigid motions do not change shape or size, which means the image of a rigid motion is always congruent to the pre-image.

When solving harder problems, should you try something even if it doesn't get you the answer? Why or why not?

Yes, you should try something instead of letting yourself get stuck because even if what you try is incorrect, or does not give you the full answer you want, it can reveal more about the problem's solution than you would have seen just looking at the problem in its original form. These new observations are likely the key to solving the problem completely.

Is a sequence of rigid motions also a rigid motion?

Yes. Each rigid motion preserves shape and size. Therefore, any number of rigid motions will also preserve shape and size.

map

a function that takes a point as an input and produces a point as an output

True or false: When a figure on a coordinate grid is translated down and to the right, the vertex with the largest y-coordinate in the pre-image will correspond to the vertex with the largest y-coordinate in the image.

This statement is true.

A rotation by how many degrees represents three full rotations?

Three full rotations is 1080°.

Translate a triangle with vertices A(−3,1),B(1,−4) and C(−4,−2)by the vector (4,4) Draw the pre-image and image on a coordinate plane, and label all vertices (The image vertices should be labeled A′A', B′B', and C′, and all vertices should be labeled with their coordinates.)

Translating by (4,4) is the same as adding 4 to each coordinate of the pre-image. A=-3,1→-3+4,1+4=1,5=A'B=1,-4→1+4,-4+4=5,0=B'C=-4,-2→-4+4,-2+4=0,2=C'

Name the transformations that you know are rigid motions, and explain why they are referred to as such.

Translations, reflections, and rotations are rigid motions. They are called rigid motions, because they move or reorient a pre-image without changing the shape and size of the pre-image. That is, the rigid motions move a figure while keeping the figure rigid.

True or false? Changing the order of a sequence of rigid motions can change the resulting rigid motion.

True

translations

When an object slides a given distance in a given direction

Is a 1440° rotation a full-circle rotation?

Yes, because 1440°=360° it represents 4 full rotations.

Does rotating by 1800° carry a figure onto itself?

Yes, because 1800° is a multiple of 360°.

Emily rotates a figure 90° four times and obtains the original figure in the same location. Do you agree with Emily's findings?

Yes, because 4×90°=360°, which is equal to a full rotation, the four rotations should result in the original figure.

Are individual points allowed to move when a figure is mapped onto itself?

Yes, the individual points can move, as in a reflection that carries a figure onto itself, but essentially every point is carried to a different location on the same figure, so the image is still the pre-image.