transformations -...

Geometry – Transformations ~1~ NJCTL.org

Transformations

Transformations: CLASSWORK Tell whether the transformation appears to be a rigid motion. Explain. 1. 2. Preimage Image Preimage Image 3. Identify the type of transformation. What is the image of segment FG?

4. Identify the type of transformation. What is the image of angle J?


Transformations: HOMEWORK Tell whether the transformation appears to be a rigid motion. Explain. 1. 2. Preimage Image Preimage Image 3. Identify the type of transformation. What is the image of segment QR?

4. Identify the type of transformation. What is the image of angle T?


Translations: CLASSWORK

Graph the image of each figure under the given translation.

1. T<–1, 4> (ΔABC) 2. T<3, -3> (JKLM)

3. Write a rule to describe the translation.

4. If T<10, 7> (QRST) = Q′R′S′T’, what translation maps ′Q′R′S′T’ onto QRST?


5. ΔXYZ has coordinates X(2, 3), Y(1, 4), and Z(8, 9). A translation maps X to X′(4, 8). What are the coordinates for Y′ and Z′ for this translation?

6. Write three different translation rules for which the image of ΔABC has a vertex at the origin.

Translations: Homework

Graph the image of each figure under the given translation. 1. T<3, -2> (ΔDEF) 2. T<-4, 0> (WXYZ)


3. Write a rule to describe the translation.

4. If T<-4, 9> (ΔLMN) = ΔL’M’N’, what translation maps ΔL’M’N’ onto ΔLMN?

5. ΔABC has coordinates A(2, 3), B(4, –2), and C(3, 0). After a translation the coordinates of A′ are (6, –2). What are the coordinates of B′ and C′?

6. Write three different translation rules for which the image of ΔBCD has a vertex at the origin.


Reflections: CLASSWORK For numbers 1-6 find the coordinates of each image. 1. Rx-axis (A) 2. Ry-axis (B) 3. Ry = x (C) 4. Rx = 2 (D) 5. Ry= -1 (E) 6. Rx = -3 (F)

7. Given points A(3, 3), B(5, -2), and C(4, 4), graph ΔABC and its reflection image as indicated. a. Ry-axis (ΔABC) b. Rx-axis (ΔABC)


c. Rx = 1(ΔABC) d. Ry = –2(ΔABC)

Draw the line of reflection you can use to map one figure onto the other.

8. 9.

10. Find the image of J(-2, 1) after two reflections, first across line ℓ1, and then across line ℓ2.

a. ℓ1 : x = 2, ℓ2 : y-axis

b. ℓ1 : y = –1, ℓ2 : y-axis

c. ℓ1 : y = 2, ℓ2 : x-axis

d. ℓ1 : x = –1, ℓ2 : y = –3

11. Give an example of a place you may see a geometric reflection in everyday life. Explain.


Identify all of the lines of symmetry for the given shape. 12. 13. 14. 15. 16. 17.

Reflections: HOMEWORK

For numbers 1-6 find the coordinates of each image. 1. Rx-axis (A) 2. Ry-axis (B) 3. Ry = x (C) 4. Rx = 3 (D) 5. Ry= 2 (E) 6. Rx = -1 (F)


7. Given points D(-2, 1), E(1, 3), and F(2, -2), graph ΔDEF and its reflection image as indicated. a. Ry-axis (ΔDEF) b. Rx-axis (ΔDEF)

c. Rx = 2(ΔDEF) d. Ry = –1(ΔDEF)


Draw the line of reflection you can use to map one figure onto the other.

8. 9.

10. Find the image of K(4, -3) after two reflections, first across line ℓ1, and then across line ℓ2.

a. ℓ1 : x = 2, ℓ2 : y-axis

b. ℓ1 : y = –1, ℓ2 : y-axis

c. ℓ1 : y = 2, ℓ2 : x-axis

d. ℓ1 : x = –1, ℓ2 : y = –3

11. Give an example of a place you may see a geometric reflection in everyday life (Different than your classwork answer). Explain Identify all of the lines of symmetry for the given shape. 12. 13. 14. 15. 16. 17.


Rotations: CLASSWORK Draw the image of each figure for the given rotation about P. Use prime notation to label the vertices of the image.

1. r(-60°, P)(∆ABC)

2. r(85°, P)(∆ABC)

3. Point O is the center of regular pentagon JKLMN. Find the image of the given point or segment for the given rotation. (counterclockwise)

a. r(144°, O)(K)

b. r(72°, O)(N)

c. r(216°, O)(ML)

d. r(360°, O)(JN)

e. r(288°, O)(JO)


4. ΔABC has vertices A(2, 2), B(3, – 2), and C(–1, 3). a. Graph r(270º, O)(∆ABC). b. Graph r(180º, O)(∆ABC).

c. Graph r(90º, O)(∆ABC).

5. The vertices of ABCD have coordinates A(2, 6), B(-3, 4), C(-3, -5), and

D(4, -2). What are the coordinates of the vertices of r (180º, O)(ABCD)?


6. The vertices of r(270º, O) (DEFG) have coordinates D’(4, 5), E’(4, -3), F’(-2, -3), and G’(-2, 5). What are the coordinates of the vertices of DEFG?

7. ABCD has vertices A(4, 2), B(–2, 2), C(–4, –2), and D(2, –2). Which of the following quadrilaterals is

r(180º, O)(ABCD)?

a. ABCD b. BCDA c. CDAB d. DABC

8. ∆FGH has vertices F(–1, 2), G(0, 0), and H(3, – 1). What are the coordinates of the vertices of r(–

270º, G)(∆FGH)? Do the following figures have rotational symmetry? If yes, what is the degree of rotation? 9. 10. 11. Rotations: HOMEWORK Draw the image of each figure for the given rotation about P. Use prime notation to label the vertices of the image. 1. r(-90°, P)(∆ABC)


2. r(110°, P)(∆ABC)

3. Point O is the center of regular hexagon ABCDEF. Find the image of the given point or segment for the given rotation. (counterclockwise)

a. r(120°, O)(F)

b. r(180°, O)(B)

c. r(300°, O)(BC)

d. r(360°, O)(FE)

e. r(60°, O)(E)

f. r(240°, O)(AB) 4. Quadrilateral DEFG has vertices D(2, 3), E(4, -2), F(7, -2), and G(5, 2). a. Graph r(270º, O)(DEFG). b. Graph r(180º, O)(DEFG).


c. Graph r(90º, O)(∆DEFG).

5. The vertices of PQRS have coordinates P(–1, 5), Q(3, 4), R(2, – 4), and S(–3, – 2). What are the

coordinates of the vertices of r (90º, O)(PQRS)?

6. The vertices of r(270º, O) (KLMN) have coordinates K’(–3, 2), L’(2, 3), M’(4, – 2), and N’(–2, – 4). What are the coordinates of the vertices of KLMN?

7. The vertices of quadrilateral ABCD have coordinates A(4, 3), B(–3, 4), C(–4, – 3), and D(3, – 4). Explain how the transformation r(90º, O)(ABCD) = BCDA can be used to show that the quadrilateral is square.


8. ∆RST has vertices at R(0, 3), S(4, 0), and T(0, 0). What are the coordinates of the vertices of r(-

270º,T)(∆RST)?

Do the following figures have rotational symmetry? If yes, what is the degree of rotation?

9. 10. 11.


Compositions of Transformations: CLASSWORK

Find the image of each letter after the transformation Rm º Rℓ. Is the resulting transformation a translation or a rotation? For a translation, describe the distance and direction. For a rotation, tell the center of rotation and the angle of rotation.

1. l 2.

K m

For numbers 3 – 6 Graph ∆ABC and its glide reflection image. A(-3, 3), B(0, 1) and C(-1,-4)

3. (Ry-axis ○ T<3, 0>)(∆ABC) 4.(Ry=1 ○ T<–1, 0>)(∆ABC)

J

l 60º m P


5. (Rx=2 ○ T<0, 1>)(∆ABC) 6. (Ry=x ○ T<3, 3>)(∆ABC)

7. Lines ℓ and m intersect at point P and are perpendicular. If a point Q is reflected across ℓ and then across m, what transformation rule describes this composition? 8. Graph AB and its image A’B’ after a reflection first across ℓ1 and then across ℓ2. Is the resulting transformation a translation or a rotation? For a translation, describe the direction and distance. For a rotation, tell the center of rotation and the angle of rotation.

A(–3, 4), B(–1, 0); ℓ1 : x = 1; ℓ2 : y = –1


9. P maps to P′(2, 3) by the given glide reflection. What are the coordinates of P? (Rx-axis º T <2, 1>)(P) Compositions of Transformations: HOMEWORK

Find the image of each letter after the transformation Rm º Rℓ. Is the resulting transformation a translation or a rotation? For a translation, describe the distance and direction. For a rotation, tell the center of rotation and the angle of rotation.

1. 2.

For numbers 3 – 6 Graph ∆DEF and its glide reflection image. D(0, 2), E(2, 4) and F(4,0)

3. (Rx-axis ○ T<2, 1>)(∆DEF) 4. (Ry=2 ○ T<–3, 1>)(∆DEF)

m

A l K

N

l

m


5. (Rx=1 ○ T<0, -2>)(∆DEF) 6.. (Ry=x ○ T<2, 3>)(∆DEF)

7. A triangle is reflected across line ℓ and then across line m. If this composition of reflections is a translation, what is true about m and ℓ?

8. Graph AB and its image A’B’ after a reflection first across ℓ1 and then across ℓ2. Is the resulting transformation a translation or a rotation? For a translation, describe the direction and distance. For a rotation, tell the center of rotation and the angle of rotation. A(–5, 2), B(–3, 6); ℓ1 : x = –2; ℓ2 : x = 3


9. P maps to P′(2, 3) by the given glide reflection. What are the coordinates of P? (Ry = x º T <3, 3>)(P) Congruence Transformations: CLASSWORK ∆ABC ∆DEF. What is a congruence transformation that maps ∆ABC to ∆DEF? 1. 2.

Identify a pair of congruent figures. Then determine a congruence transformation that maps the preimage to the congruent image.

3. 4.


Congruence Transformations: HOMEWORK ∆ABC ∆DEF. What is a congruence transformation that maps ∆ABC to ∆DEF? 1. 2.

Identify a pair of congruent figures. Then determine a congruence transformation that maps the preimage to the congruent image. 3. 4.


Dilations: CLASSWORK A dilation has center (0, 0). Find the image of each point for the given scale factor. 1. A(3, 4); D7(A) 2. B(0, 4); D3.4(B) 3. C(5, –6); D5/3(C)

4. A square has 16-cm sides. Describe its image for a dilation with center at one of the vertices and scale factor 0.8.

5. Graph quadrilateral ABCD and its image A′B′C′D’ for a dilation with center (0, 0) and a scale factor

of 2. The vertices of ABCD are: A(2, 3), B(-2, 4), C(-3, -2), D(3, –3).


The solid-line figure is a dilation of the dashed-line figure. The labeled point is the center of dilation. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor of the dilation. 6. 7.

8. D(2, K)(∆ABC)

9. ∆ABC is dilated in the coordinate plane with center (2, -1) and scale factor 0.5. If A(6, 3) what is the y-coordinate of A’ ? Dilations: HOMEWORK A dilation has center (0, 0). Find the image of each point for the given scale factor. 1. P(–3, 5); D1. 2(P) 2. O(–2, –1); D4(O)


3. M(4, 2); D3/2(M) 4. A square has 12-cm sides. Describe its image for a dilation with center at one of the vertices and scale factor 0.8. 5. Graph pentagon JKLMN and its image J′K′L′M′N′ for a dilation with center (0, 0) and a scale factor

of 1.5. The vertices of JKLMN are: J(0, 3), K(3, 3), L(3, 0), M(0, –3), N(–1, 0).


The dashed-line figure is a dilation of the solid-line figure. The labeled point is the center of dilation. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor of the dilation. 6. 7.

8. D(3, A)(∆ABC)

9. ∆DEFis dilated in the coordinate plane with center (-5, 2) and scale factor 2. If B(-5, 4) what is the y-coordinate of B’ ?


Similarity Transformations: CLASSWORK ∆JKM has vertices J(0, 4), K(4, 4) and M(0, 3). Draw the image for each similarity transformation. 1. (D2

Rx-axis) (∆JKM) 2. (T<-3, -2> D1.5 ) (∆JKM)

3. (D0.5

r(270º, O)) (∆JKM)


4. For the graph, describe the composition of transformations that maps ΔDEF to ΔLMN.

5. Show that ΔABC~ΔFGH by finding a similarity transformation that maps one triangle to the other.

6. Determine if there is a similarity transformation that maps one figure onto the other. If so, identify the similarity transformation and write a similarity statement. If not, explain.

7. Do the compositions T<4, – 2> ○ D2 and D2 ○ T<2, – 1> describe the same similarity transformation? Explain.


Similarity Transformations: HOMEWORK ∆JKM has vertices J(-2, 2), K(2, 0) and M(1, -2). Draw the image for each similarity transformation. 1. (D2

Ry-axis) (∆JKM) 2. (D0.5 T<2, -2>) (∆JKM)

3. (r(-90º, O) D1.5 )(∆JKM)


4. For the graph, describe the composition of transformations that maps ΔDEF to ΔLMN.

5. Show that ΔABC~ΔFGH by finding a similarity transformation that maps one triangle to the other. 6. Determine if there is a similarity transformation that maps one figure onto the other. If so, identify the similarity transformation and write a similarity statement. If not, explain.

7. A similarity transformation is described by the composition T<2, 1> ○ D3. If the order of the

composition is changed to be D3 ○ T<2, 1>,does that describe the same transformation? Explain.


ANSWER KEY

Transformations: CLASSWORK 1. Yes, distances and angle measures are preserved. 2. No, distances are not preserved. 3. Rotation; F’G’ 4. Dilation; angle J’

Transformations: HOMEWORK 1. No, distances are not preserved. 2. Yes, distances and angle measures are preserved. 3. Reflection; segment G’R’ 4.Translation; angle T’ Translations: CLASSWORK

1. 2.


3. T<10, -1> (ABDC) 4. T<-10, -7> (Q’R’S’T’) = QRST 5. Y’(3, 9); Z’(10, 14) 6. T<-2, -2> (ΔABC); T<-2, 6> (ΔABC); T<-5, -3> (ΔABC) Translations: HOMEWORK 1.

2.


3. T<-4, 10> (ΔIJK) 4.T<4, -9> (ΔL’M’N’) = ΔLMN 5. B’(8, -7); C’(7, -5) 6. T<-4, -2> (ΔBCD); T<-4, -9> (ΔBCD); T<-8, -6> (ΔBCD) Reflections: CLASSWORK 1. A’(1, -3) 2. B’(-5, 5) 3. C’(-1, 3) 4. D’(7, 1) 5. E’(-2, 1) 6. F’(-3, 4) 7. a. A’(-3, 3), B’(-5, -2), C’(-4, 4) b. A’(3, -3), B’(5, 2), C’(4, -4)

c. A’(-1, 3), B’(-3, -2), C’(-2, 4) d. A’(3, -7), B’(5, -2), C’(4, -8)


8. 9.

10. a. J’(-6, 1) b. J’(2, -3) c. J’(-2, 1) d. J’(0, -7) 11. Answers may vary. Sample answer: Patterns in tiles, wallpaper, etc.

Reflections: HOMEWORK 1. A’(5, -1) 2. B’(-3, 2)


3. C’(-2, 1) 4. D’(12, 1) 5. E’(-1, 4) 6. F’(0, -5) 7. a. D’(2, 1), E’(-1, 3), F’(-2, -2) b. D’(-2, -1), E’(1, -3), F’(2, 2)

c. D’(6, 1), E’(3, 3), F’(2, -2) d. D’(-2, -3), E’(1, -5), F’(2, 0)

8. 9.


10. a. K’(0, -3) b. K’(-4, 1) c. K’(4, -7) d. K’(-6, -3) 11. Answers may vary. Sample answer: Patterns in tiles, wallpaper, etc.

Rotations: CLASSWORK 1.

2.


3. a. N b. M c. NJ d. JN e. KO 4. a. A’(-2, 2), B’(2, 3), C’(-3, -1) b. A’(-2, -2), B’(-3, 2), C’(1, -3)

c. A’(2, -2), B’(-2, -3), C’(3, 1)

5. A’(-2, 6), B’(3, -4), C’(3, 5), D’(-4, 2) 6. D’(5, -4), E’(-3, -4), F’(-3, 2), G’(5, 2) 7. C


8. F’(-2, -1), G’(0, 0), H’(1, 3) 9. Yes, 180º 10. Yes, 90º 11. Yes, 180º Rotations: HOMEWORK 1.

2.

3. a. D b. E c. DC d. FE e. D f. DC 4. a. D’(-3, 2), E’(2, 4), F’(2, 7), G’ (-2, 5) b. D’(-2, -3), E’(-4, 2), F’(-7, 2),G’(-5, -2)


c. D’(3, -2), E’(-2, -4), F’(-2, -7),G’(2, -5)

5. P’(5, 1), Q’(4, -3), R’(-4, -2), S’(-2, 3) 6. K(2, 3), L(3, -2), M(-2, -4), N(-4, 2) 7. Answers may vary. Sample: A rotation is a rigid motion therefore distances and angle measures are preserved. AB = BC = CD = AD and m<A = m<B = m<C = m<D. If all angles of a quadrilateral are equal then they each must measure 90 degrees. By definition the quadrilateral is a square. 8. R’(-3, 0), S’(0, 4), T’(0, 0) 9. Yes, 45º 10. Yes, 22.5º 11. Yes, 180º Compositions of Transformations: CLASSWORK 1. translation, down and to the right, twice the distance between lines l and m 2. rotation, about point P, 120º clockwise


3. A’(0, 3), B’ (-3, 1), C’(-2, -4) 4. A’(-4, -1), B’ (-1, 1), C’(-2, 6)

5. A’(7, 4), B’ (4, 2), C’(5, -3) 6. A’(6, 0), B’ (4, 3), C’(-1, 2)

7. 180º rotation 8. A rotation, about point (1, -1), 180º clockwise


9. P(0, -4) Compositions of Transformations: HOMEWORK 1. rotation, about point K, 180º counterclockwise 2. translation, directly down, twice the distance between lines l and m 3. D’(2, -3), E’ (4, -5), F’(6, -1) 4. D’(-3, 1), E’ (-1, -1), F’(1, 3)

5. D’(2, 0), E’ (0, 2), F’(-2, -2) 6. D’(5, 2), E’ (7, 4), F’(3, 6)


7. m || l 8. A translation, 10 units to the right

9. P(0, -1) Congruence Transformations: CLASSWORK


1. Check student work. Sample Answer: (T <1, 0>º Ry -axis º r(270º, O))(ΔABC) = ΔDEF 2. Check student work. Sample Answer: (T <2, 0>º Ry -axis º r(180º, O))(ΔABC) = ΔDEF

3. ΔABC≅ΔGHI; Check student work. Sample Answer: r(180º, O))(ΔABC) = ΔGHI 4. ABCD≅LIJK; Check student work. Sample Answer: T <-5, -4>(ABCD)≅LIJK Congruence Transformations: HOMEWORK 1. Check student work. Sample Answer: (Rx -axis º T <5, 0> (ΔABC) = ΔDEF 2. Check student work. Sample Answer: (r(90º, O) º Ry -axis )(ΔABC) = ΔDEF

3. CD≅EF; Check student work. Sample Answer: Ry -axis (CD) = EF

4. ΔABC≅ΔIHG; Check student work. Sample Answer: Ry =2 ºT <6, 0> (ΔABC) = ΔIHG

Dilations: CLASSWORK 1. A’(21, 28) 2. B’(0, 13.6) 3. C’(25/3, -10) 4. A square with 12.8 cm sides. 5. A’ (4, 6), B’(-4, 8), C’(-6, -4), D’(6, -6)

6. scale factor: 3; enlargement 7. scale factor: 1/4; reduction 8.


9. 1 Dilations: HOMEWORK 1. A’(-3.6, 6.0) 2. B’(-8, -4) 3. C’(6, 3) 4. A square with 9.6 cm sides. 5. J’ (0, 4.5), K’(4.5, 4.5), L’(4.5, 0), M’(0, -4.5), N’(-1.5, 0)

6. scale factor: 9/5; enlargement 7. scale factor: 1/3; reduction 8.

9. 6


Similarity Transformations: CLASSWORK 1. J’(0, -8), K’(8, -8), M’(0,6)

2. J’(-3, 4), K’(3, 4), M’(-3, 2.5)


3. J’(-2, 0), K’(-2, 2), M’(-1.5, 0)

4. Answers may vary. Sample: (D0.5 ºr(180 ,O))(ΔDEF) = ΔLMN 5. Answers may vary. Sample: (T<-4, -10>º D3)(ΔABC) = ΔFGH 6. No, the corresponding sides are not proportional. 7. Yes. For a point (x, y), the first transformation will transform (x, y) to (2x + 4, 2y-2) while the second transformation will transform it to (2(x+2), 2(y-1)) which = (2x + 4, 2y-2) Similarity Transformations: HOMEWORK 1. J’(4, 4), K’(-4, 0), M’(-2, -4)


2. J’(0, 0), K’(2, -1), M’(1.5, -2)

3. J’(-3, -3), K’(0, 3), M’(3, 1.5)

4. Answers may vary. Sample: (Rx -axis º D2)(ΔDEF) = ΔLMN 5. Answers may vary. Sample: (r(90 ,O) º D1.5 )(ΔABC) = ΔFGH 6. Yes. A reflection followed by a dilation with a scale factor of 0.5. ΔABC~ΔEDC 7. No. For a point (x, y), the first transformation will transform (x, y) to (3x + 2, 3y+1) while the second transformation will transform it to (3(x+2), 3(y+1)).

transformations -...

Documents