Rigid Motions HW packet

Name: ___________________________________

Compositions of Transformations The transformations you have learned so far are:

Name Meaning Notation

Translations add the specified amount to the coordinates of each point. T3,4

Reflect in x change the sign of the y-coordinate. rx-axis

Reflect in y change the sign of the x-coordinate ry-axis Reflect in y=x switch the coordinates of each point ry=x Reflect in y=-x switch and negate the coordinates of each point. ry=-x Rotate 90o switch the order and then change the sign of the x-coordinate. R90

Rotate 180o Apply the 90o rotation twice R180 Rotate 270 o Apply the 90o rotation three times R270

Use the rules to solve the problems below.

1. A(-3, 5) T(2 ,1) _____________

2. B(-2, -4) T(2 ,1) _____________

3. C(3, -5) rxaxis _____________

4. D(5, 3) ryaxis _____________

5. E(-3, -2) ryx _____________

6. R(6, 2)

R90 ______________

7. S(-5, 3)

R180 ___________

8. T(7, -1)

R270 ____________

9. __________ryx F’(7, -2)

10. __________ryaxis G’(3, -9)

11. __________T(3,5 ) J’(7, 3)

12. __________rxaxis K’(-3, 2)

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A shape can be transformed more than once. Such a transformation is called a composition of transformations. The transformations are written together with a small between them.

5,12Trx means translate first and then reflect.

yaxisrT )4,2( means _____________________________ first and then __________________.

360Rr axisy means _____________________________ first and then __________________.

2,390 TR means _____________________________ first and then __________________.

Now answer the following. Show all your work.

1. What is the image of A(-3, 5) under 5,190 TR ?

2. What is the image of P(-3, -9) under 1)4,2( yrT ?

3. What is the image of S(3, 6) under

R180 T(1, 2) ?

4. What is the image of Q(-2, 8) under 1 yaxisy rr ?

5. Graph ABC with A(3, 5), B(5,3) and

C(3, -1). Graph the image of ABC under

3,5 Tr axisx

6. Graph PQR with P(2, -1), Q(2,1) and

R(-2, 0). Graph the image of PQR under

4,4 Tr axisy .

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7. Graph DEF with D(-5, 5), E(-8,2) and

F(-3, 0). Graph the image of DEF under

T2,1 R90 .

8. Graph ABC with A(8, 1), B(6,3) and

C(5, -1). Graph the image of ABC under

R180 T3,1.

9. Graph DEF with D(4,4), E(6,2) and

F(2,0). Graph the image of DEF under

180Rr axisx .

10. Graph DEF with D(-1,-5), E(-5,-4) and

F(-3,-1). Graph the image of DEF under

T3,2 R90 .

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11. Graph DEF with D(1,1), E(5,1) and

F(3,3). Graph the image of DEF

under

ryx R90 .

12. Graph ABC with A(-1,1), B(-4,3) and

C(-2, 0). Graph the image of ABC under

0,2180 TR .

13. Graph DEF with D(1,-1), E(3,-1) and

F(2,-4). Graph the image of DEF under 3 xaxisy rr .

14. Graph DEF with D(3,3), E(9,6) and

F(6,9). Graph the image of DEF under 3601,4 RT .

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Transform-

ation

Figure Are Corresponding Parts

Congruent?

(is it a rigid motion?)

Segments between

Corresponding Points

Other Relationships

Coordinate Rules

Rotation

Reflection

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Transform -

ation

Figure Are Corresponding

Parts Congruent?

(is it a rigid motion?)

Segments between

Corresponding Points

Congruent?

Other Relationships

Coordinate Rules

Translation

Dilation

Mapping a regular polygon to itself:

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Transformations Practice #1

Multiple Choice 1. Which of the following transformations always preserves slope?

(1) translation (3) reflection (2) rotation (4) glide reflection

2. In the accompanying figure, ∆A'B'C' is the image of ∆ABC after a translation. Which of the following is not necessarily true?

(1) (3)

(2) // (4) 3. In the accompanying figure, ∆DFE is the image after RO, 80(∆ABC). Which if the following must be true? (1) mA 80 (3) m A m C (2) mBOF = 80 (4) mABE = 80

4. is the reflection of over line m. Which if the following is not necessarily true? (1) the midpoint of GE lies on line m (2) GE = OW

(3) line m (4) 5. ∆MNP is the reflection of ∆ZXY through point O. Which of the following must be true? (1) YM = XN (2) YZ = ZX (3) MN = YZ (4) YZ = MP

OA B

C

D E

F

W

E

G

O

m

OX

Y

Z

M

N

P

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6. Pentagon ABCDE is the image of pentagon VWXYZ after DC,3. Which of the following must be true? (1) VW = 3AB (3) CV = 3VA (2) CV = 3CA (4) mA = 3(mV)

7. Given RO,110( ) = . If mHOG = 45, what is the measure of GOP? (1) 25 (2) 45 (3) 65 (4) 70

8. Which transformation maps ∆ABC onto ∆A'B'C'? (1) RA,90 (2) RB,90 (3) (4) ⃗⃗⃗⃗⃗⃗ ⃗

9. Given the following angle measures, which point could be the center of rotation that maps ∆ABC onto ∆A'B'C'?

(1) X (2) Y (3) Z (4) neither X, Y, or Z could be the center 10. In regular hexagon ABCDEF, which transformations would map ∆AQF to ∆DQC? (1) ⃗⃗⃗⃗⃗⃗

(2) reflect through point Q (3)

(4) RQ,60

A B

C

A'

B'C'

m AXA' = 30

m CXC' = 30

m CYC' = 35

m BYB' = 40

m BZB' = 45

m CZC' = 48

o

o

o

o

o

o

XY

Z

G

H

P

Q

O

A B

C

B'

C'

A

B C

D

EF

Q

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11 In the figure to the right RPS measures 30, and is rotated 160about point P. R'PS' is the

image after the rotation. Find the measure of SPR'. 12. Describe fully the transformation that maps ∆XYZ to ∆XY'Z'. 13. When an equilateral triangle is reflected over one of its sides, what type of figure is formed? Sketch an example. 14. Given point Y is the image of point X after a

reflection over , explain why PXZ PYZ? 15. Estimate the location of the center of rotation. Use a protractor to determine how close your estimate was by measuring the angle formed by two pairs of corresponding points and your center.

X Y

Z

P

A

B D

E

F

CA'

B'

C'

D'

E'

F'

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16. Sketch segment and midpoint P on . Sketch the rotation of about point P by 90.

Repeat for point C that lies on but is not the midpoint. 17. Sketch the line of reflection of the following figure: 18. Sketch ⃗⃗ ⃗⃗ ⃗(P)

21. Sketch the dilation of the triangle through point O with a scale factor of 2. 22. Shaina states that whenever a segment is reflected over a line, the two endpoints of the image and the two endpoints of a pre-image form a trapezoid. Is she correct? Justify your reasoning.

P

A

B

T T

O

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Transformations Practice#2 1. Which letter has rotational symmetry?

(1) T (2) A (3) Z (d ) E

2. What is the smallest angle of rotation that would map a hexagon onto itself? (1) 45 (2) 60 (3) 120 (4) 180 3. Which of the following figures has rotational symmetry? (1) equilateral triangle (2) scalene triangle (3) obtuse triangle (4) right triangle 4. Which of the following figures have exactly two lines of symmetry?

(1) 2) 3) 4) 5. By how many degrees must the octagon be rotated to map point X to point Y?

(1) 45 (2) 90 (3) 135 (4) 225 6. By how many degrees must the decagon be rotated to map point R to point S? (1) 36 (2) 144 (3) 216 (4) 324

X

Y

R

S

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7. Which word has a horizontal line of symmetry?

(1) HIDE (2) BOSS (3) BAT (4) TOOT

8. If pentagon GHIJK is rotated 288 about its center P, then vertex G maps to vertex H. What is the measure of the acute angle GPQ? (1) 36 (2) 72 (3) 84 (4) 88 9. Sketch all lines of symmetry in the following figure. 10. Danny and Sue are riding a Ferris wheel at a carnival. The Ferris wheel has 20 cars and spins at a rate of 6 degrees per seconds. Danny and Sue are in separate cars that are three positions apart. When Danny is at the top of the Ferris wheel, how many seconds later will Sue be at the top? 11. Sketch an example of a figure that has one line of symmetry but does not have rotational symmetry, 12. Jessie states that any triangle with line symmetry must be equilateral, but Fiona disagrees. Who is correct? Justify your answer.

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Transformations Practice #3 1. Which composition could result in (12,4) mapping to (5,1) ? (1) D1/4 ry=x (3) R90 T9,-11 (2) D1/4 T2,0 (4) T9,-11 R90 2. Which composition of transformations would not fit the two puzzle pieces together

(1) T

GA u ruuu or

GFHEFG( )

(2) T

FA u ruuu or

GFHEFG( )

(3) r

HEo T

HBu ruu HEFG( )

(4) T

HB u ruuu or

HEHEFG( )

3. Which composition could result in the following pre-image and image?

(1) R180°o D

1/2 (2) D2

o R180° (3)

r

y-axiso D

1/2 (4)

D

1/2oT

5,0

4. Which composition could result in (5,1) (1,3) (1) T-6,2(rx-axis) (2) rx-axis(T6,2) (3) R90(T2,2) (4) T6,2(ry-axis) 5. Which image represents reflection following by a translation parallel to the line of reflection?

(1) (2) (3) (4) 6. Rayquan is designing a website logo using a computer graphics program. He uses the programs enlarge feature to double the logo’s size, and then drags it horizontally 2 inches right and vertically 3 inches up. Which composition represents the change in the logo?

(a) (b) (c) (d)

A B

CD

E F

GH

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(1) T

2,3o D

1/2 (2)

T

2,3o D

2

(3) T

3,2o D

1/2 (4)

D

2oT

3,2

7. When equilateral triangle ∆ABC undergoes the

transformation R

B,240°o r

AB, which corresponding

pairs of vertices coincide? (1) A, A'' and B, B'' only (2) B, B'' and C, C'' only (3) C, C'' and A, A'' only (4) A, A'', B, B'' and C, C''

8. The figure represents r

line sR

C ,120°AB( )( ). Sketch line s and point C in the figure.

9. The vertices of ∆BIG have coordinates B(6, 1) I(3, 2) and G(7, 5). Graph and label ∆B''I''G'', the

image of ∆BIG after R

y=xo R

origin,90°. State the coordinates.

10. The vertices of ∆PMN have coordinates P(4, 0) M(3, 3) and N(6, 5). Graph and label

∆P''M''N'', the image of ∆PMN after T

-4,1o R

x=3. State the coordinates.

A

B

A'

B'

A''

B''

A B

C

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Congruence with rigid motions

1. In the figure below, ∆DAB ∆BED ∆EBC ∆CFE. Which of the following sequences of

rigid motions will always map ∆DAB to ∆CFE?

(1) a translation by vector BE followed by a translation by vector EC

(2) a rotation of 180 about point B, followed by a translation by vector CF

(3) a reflection over BE followed by a translation by vector BC

2. Triangles ∆AMN and ∆APQ share vertex A. A circle can be constructed that has a center

at A and passes through points M, N, P, and Q. If mMAN = mPAQ, Which of the

following is not necessarily true??

(1) NAQ is a right angle (2) M Q

(3) ∆APQ is the image of ∆AMN after a rotation about point A. (4) ∆AMN and ∆AQP are

isosceles

3. Regular hexagon ABCDEF is shown in the figure below. Which of the following

transformations will map the hexagon onto itself such that point D maps to point E?

(1) a 60 rotation bout point P followed by a reflection of line BE

(2) reflection over line BE followed by 60 rotation about point P

(3) reflection over lines CF followed by a translation by vector FE

(4) reflection over lines CF followed by a 120 rotation about point P

A B C

D E F

A

M N

P

Q

PA

B C

D

EF

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4. Which of the following will map parallelogram ABCD onto itself?

(1) A rotation of 180 about the point of intersection of the two diagonals of the ABCD

(2) A reflection over CD , followed by a translation along vector DA

(3) A rotation of 180 about point B, followed by a translation along vector BA

(4) A reflection over BC , followed by a translation a rotation of 180 about point B

5. Given right triangles ABC and DEF where C and F are right angles, AC DF and CB

FE . Describe a precise sequence of rigid motions which would show ∆ABC ∆DEF.

6.

A B

CD

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