Math B Assignments: Introduction to Proofs



Geometry Assignments: Transformations

|Day |Topics |Homework |HW Grade |Quiz Grade |

|1 |Line reflections |HW TG - 1 | | |

|2 |Rotations |HW TG - 2 | | |

|3 |Translations |HW TG - 3 | | |

|4 |Practice **QUIZ** |HW TG - 4 | | |

|5 |Dilations |HW TG - 5 | | |

|6 |Compositions |HW TG - 6 | | |

|7 |Practice **QUIZ** |HW TG - 7 | | |

|8 |Rigid motions |HW TG - 8 | | |

|9 |Rigid Motions and Congruence |HW TG - 9 | | |

|10 |Review **QUIZ** |HW TG - Review | | |

|11 |**TEST** | | | |

Geometry: Transformations Answers

HW - 1

1b. Construct the perpendicular bisectors of the segments joining each vertex to its image (the ( bis. of [pic], [pic], and[pic]). If they are all the same line, it’s a reflection.

3a. (–2, –7) b. (–7, 2) c. (2, 7) 4. (–3, 1) 5. (4, 5) 6. x = 2

7a. B b. (0, 3) and (0, 4) c. (2, 2) and (4, 4)

8a. x = 5 b. y = x – 2 c. y = –2.5 9. a and b 10c. (–4, 1)

11. x = 1, y = 2, y = x + 1, y = –x+ 3

HW - 2

2. (–1, –3) 3a. (–4, 5) c. Nothing

4. (2, 5) 5a. P'(x, –y) b. P"(–x, –y) c. Ro(x, y) 6. a and e are rotations

7a. Clockwise; same. b. a and e are same; rest are opposite

c. A rotation preserves orientation (keeps it the same). A reflection changes orientation to the opposite.

8c. y = 0.5x – 2 d. y = –2x – 4

10a. 45( CCW b. 4; rotations preserve distance c. 90(; rotations preserve angle measure

10a. (–3, –4) b. (–4, –3) c. (–2, 0) d. (–3, 2) e. (–7, –5)

11a. 18( b. 15 (pentadecagon, another word you don’t need to memorize)

HW - 3

1. (5, 1) 2. (3, –4) 3. T2, –3(x, y) or (x + 2, y – 3) 4a. (0, 2) b. (1, –1)

5. A'(1, 0), B'(3, 3), C'(6, –1)

6a. 5 b. 3/4 7d. T6, –2 8d. T6, 0 9. (x + a, y + b) b. b/a c. [pic]

9a. Reflection over [pic] b. 180( rotation around M c. Reflection over ( bisector of [pic]

d. –90( rotation around M e. Translation along vector from A to C f. Reflection over ( bisector of [pic]

HW – 4

2a. a = 2, b = –1 3a. 180( around midpoint of [pic] b. Reflection over ( bisector of [pic]

4a. n b. 360/n 5. c and d are not preserved 6. c is not preserved

7a. [pic] b. [pic] c. Distance is preserved in rotations

8a. (APQ ( (BPQ b. Angle measure is preserved in reflections

c. [pic] d. Distance is preserved in reflections

10. a. 1) Translate along the vector [pic] so A goes to O and C goes to D

2) Reflect over [pic] so T goes to G

b. 1) Translate along the vector [pic] so A goes to O

2) Rotate 90( CCW around O so T goes to G and C goes to D

HW - 5

1. (8, –4) 2. (–6, 15) 3. (2, –6) 4. 4 5. 2/3

6a. S b. P c. O d. [pic] e. 1/2 f. 12

7. (3) 8a. 4 b. 12 c. 6 9c. [pic], [pic] d. 3 e. 3, 27 f. 9

10a. P = 90, A = 486 b. P = 15, A = 13.5 11e. same, parallel

HW - 6

1. (–9, –12) 2. (–5, 4) 3. (2, (4) 4. ((4, –1) 5. R90( 6. D6

7a. [pic] b. [pic] c. No

8a. [pic] or [pic] or [pic] (there are others) b. [pic] c. [pic]

9. 1) Translate along vector [pic] 2) Rotate CW until [pic] coincides with [pic] 3) Reflect over [pic]

10d. Yes. When the translation is parallel to the line of reflection.

HW - 7

1. (–3, –4) 2. (1, 3) 3. (15, –6) 4. (6, –3) 5. (4, –2) 6. (4, –5) 7. (–4, –2)

8. (3, –2) 9. (8, –2) 10. (2, –1.5) 11. (2, 6) 12. (–4, 5) 13. k = –4/3

14a. D3 b. T4,8 c. (4, 8) d. [pic]

15. (–1, 2) 16. y = 4 17a. (0, 2) b. (1, –1) 18a. 45( b. 15

19. [pic] or [pic] (There are other ways.)

20. 1) Translate along vector [pic] 2) Rotate CW until [pic] aligns with [pic] 3) Reflect over [pic]

HW - 8

1a. A transformation that preserves distance: translations, (line) reflections and rotations.

b. Yes, yes, yes, yes, no, no. 2. (5) 3. (1)

4a. (O) b. (D) c. (D) d. (N) e. (O)

f. (N) g. (D) h. (D) i. (D) j. (O)

5. a, c, f and h are rigid motions. b, g. Distance and angle measure not preserved

d. Changes shape e. Lines not preserved

6a. 90( rotation CCW followed by a translation.

b. 90( rotation CW followed by vertical line reflection and (if necessary) a translation.

c. 180( rotation (either way) followed (if necessary) by a translation.

d. 45( rotation CCW followed by a horizontal line reflection and (if necessary) a translation.

Note: all of the above have other possible answers.

6a. [pic] b. [pic] c. [pic] d. [pic] e. [pic] f. [pic]

Again, other answers are possible.

HW - 9

1. They must have measure (length). 2. They must have the same measure (degrees).

3a. (DCE ( (BAF, (CED ( (AFB, (EDC ( (FBA b. [pic], [pic], [pic]

4. No (consider two rectangles of different sizes) 5. No (consider a square and a non-square rhombus)

6a. (CBM b. Reflection over line BM.

7a. (RTS b. Reflection over horizontal line followed by translation

8a. LOVE b. 180( rotation followed (if necessary) by translation

9a. RQPS b. CCW rotation ‘til [pic], then vertical line reflection, then (if necessary) translation.

10. 60 11. 120(

Review Answers

1. (2, 3) 2. (–6, 8) 3. x = 3 4. (–2, 4) 5. (4) 6. (3) 7. (1)

8. (1) 9. (1 ) 10. (2) 11. (2) 12. (4) 13. (4) 14. (1)

15. (1) 16. (3) 17. (4) 18. [pic] or [pic] 19. (6, 3)

20a. 3/4 b. 5 c. 3/4 d. 30 21a. 5/2 b. 10 c. 12

22. 6 23a. x = 1, y = 2, y = x + 1, y = –x + 3 b. 90(, 180(, 270( or 360( around (1, 2)

24a. 31.5 b. 47.25

25a. Reflect over y = 1 then translate right 4.

b. Rotate 90( CW about the origin, reflect over the x-axis, then translate down 3 OR

Rotate 90( CW about the origin, then reflect over the line y = –1.5 OR

Reflect over the line y = x then translate down 3

c. Dilate by factor of 2 centered at the origin, rotate CCW 90( about the origin, then translate left 1, up 2 OR

Rotate CCW 90( about the origin, translate left 1, up 2, then dilate by factor of 2 centered at (–1. 2)

26. 1) Translate so B maps A. 2) Rotate CW around A until I maps to T. 3) Reflect over [pic] so X maps to R and E maps to H[pic][pic]

Geometry HW: Transformations - 1

Name

1. Consider the diagram at right of YAK and Y'A'K'.

a. Suppose Y'A'K' is the image of YAK after a line reflection. Use a compass and straight edge to construct the line of reflection. HINT: Consider property 2 of a reflection.

b. Suppose we are not sure if Y'A'K' is the image of YAK after a line reflection. Describe briefly but precisely how we could find out if it is.

2. Construct [pic], the reflection of [pic] over line k.

3. Find the coordinates of the image of the point (2, –7) under each of the following:

a. ry-axis. b. ry = x. c. rx-axis.

4. If the point (3, –1) is reflected in the x-axis and then that image is reflected in the y-axis, what are the coordinates of the final image?

5. What are the coordinates of the image of the point (4, 1) after a reflection in the line y = 3?

6. The image of the point A(–3, 1) after a reflection in line k is (7, 1). Find the equation of line k.

7. Triangle ABC is shown in the graph at right.

a. Which point on the triangle will be invariant under a

reflection in the x-axis?

b. Give the coordinates of the points on the triangle that will be

invariant (unchanged) under a reflection in the y-axis.

(Invariant points are often called fixed points.)

c. Give the coordinates of the points on the triangle that will be

invariant under a reflection in the line y = x.

8. For each diagram below, (C'A'T' is the image of (CAT after a line reflection. Write the equation of the line of reflection.

a. b. c.

9. In which of the following is (A'B'C' not the image of (ABC after a reflection?

a. b. c. d.

10. a. Graph the line l having equation [pic] and the point A(8, 9).

b. A' is the image of A after a reflection over line l. What is the slope of [pic]? Why?

c. Find the coordinates of A'.

Read: Line Symmetry

A figure has line symmetry if it is its own image after a line reflection. Alternate wording (popular on the Regents): a figure “maps onto itself” after a reflection over a line of symmetry.

In the rhombus shown at right, lines ( and k are lines of symmetry; line n is not.

11. A square is graphed at right. Write equations for all the lines of symmetry for this square.

Geometry HW: Transformations - 2

Name

1. Segment AB is to be rotated counterclockwise about center P by the angle (.

a. Use a compass to locate A', the image of A.

b. Use a compass and straight edge to locate B', the image of B. (There is more than one way to do this. The easiest is to remember that rotations preserve distance. So A'B' = AB and P'B' = PB.)

2. What is the image of (–3, 1) under a rotation of 90° about the origin?

3. a. What is the image of (4, –5) under a rotation of 180° about the origin?

b. What is the difference between this and a rotation of –180( (i.e. 180( CW) about the origin?

4. Using the rule (x, y) ( (–y, x), find the image of A(5, –2).

5. a. Find the coordinates P', the image of P(x, y) after a reflection in the x-axis.

b. Find the coordinates P", the image of P' after a reflection in the y-axis.

c. A reflection in the x-axis followed by a reflection in the y-axis is the same as what single

transformation?

6. In each of the following, tell whether the shaded triangle is the image of the unshaded triangle after a reflection (r) or a rotation (R).

a. b. c. d. e. f.

7. a. Review problem #6a. Starting at A in (ABC and going in alphabetical order around the triangle, do the points go clockwise or counterclockwise? This is called the orientation of the figure. Is the orientation of the image, (A'B'C, the same or the opposite?

b. For each of 6b – 6f, determine if the orientation of the image is the same or the opposite of the original. (Note: In each figure, the original is clockwise.)

c. How can we use orientation to tell if a figure has been reflected or rotated?

8. a. Graph the line (: y = 0.5x + 2.

b. Graph line k, the image of line ( under a reflection in the origin. One way you can do this:

Pick three points on ( and find their images.

c. Write an equation for line k.

d. Graph line n, the image of line ( after a 90( rotation about the origin.

e. Write an equation for line n.

9. a. Graph ΔRAT having coordinates R(0, 2), A(2, 5) and T(7, 2).

b. Graph ΔR'A'T', the image of ΔRAT after a 90( rotation about the origin.

c. Graph ΔR"A"T", the image of ΔRAT after a reflection in the line y = x.

10. In the diagram at right, (R'A'T' is the image of (RAT after a rotation around point P.

a. What is the angle and direction of rotation? (You do not need a protractor, just your brain.)

b. What is the length of [pic] ? How do we know?

c. What is the measure of (R'T'A' ? How do we know?

Read: Rotational Symmetry and Point Symmetry

A figure has rotational symmetry if it is the image of itself (maps onto itself) after a rotation of 0( < ( < 360(.

Ex: Ex:

11. a. What is the minimum angle of rotation that will take a regular icosagon onto itself? (An icosagon is a

20-sided polygon; that is not a vocab word you need to remember.)

b. A regular polygon has a minimum angle of 24( to carry it onto itself. How many sides does it have?

Geometry HW: Transformations - 3

Name

1. Use a compass and straight edge to translate

triangle (ABC the distance and direction of

the directed line segment (aka vector) PQ.

2. Find the image of the point (3, 5) under the translation (x, y) ( (x + 2, y – 4).

3. Under a certain translation, the image of A is R. What two things must be true about segments [pic]and [pic]for us to be sure that under that same translation the image of B is S?

4. Under a given translation, the image of (4, 2) is (6, –1).

a. Find the image of (–2, 5) under the same translation.

b. Find the preimage of (3, –4) under the same translation.

5. In the diagram at right, (ABC has vertices A(–2, 2), B(0, 5) and

C(3, 1). Vector [pic] has initial point P(–1, –1) and terminal point

Q(2, –3). State the coordinates of the vertices of (A'B'C', the image of (ABC after the transformation [pic].

6. Under the translation P(x, y) ( P'(x + 4, y + 3),

a. What is the distance between any point P and its image P' ?

b. What is the slope of the line PP' ?

7. a. Graph y = x2 + 3 for –2 ( x ( 2.

b. Draw the image of the graph in part a after a translation T2, –5.

c. Draw the image of the graph in part b after a translation T4, 3.

d. Name the single transformation that is equivalent to T2, –5 followed by T4, 3.

8. In each of the following, the dashed triangle is the image of the solid triangle after a single transformation. Name the transformation. Be specific (e.g., “Reflection over line _______;” not just “reflection”).

a. b. c.

d. e. f.

Geometry HW: Transformations - 4

Name

1. a. Graph the circle C having equation [pic].

b. Graph circle C', the image of circle C after a reflection in the x-axis.

c. Graph circle C'', the image of circle C after a reflection in the line y = x.

d. Graph circle C''', the image of circle C after a 90° rotation about the origin.

e. Graph circle C*, the image of circle C after a translation along the vector [pic].

2. Given points A(2, 4) and B(6, 2). What are the values of a and b

such that the translation Ta, b maps A onto the midpoint of [pic]?

3. Let [pic] be a line segment.

a. What rotation would map [pic] onto itself? (Be specific: what point and what angle?)

b. What reflection (other than over itself) would map [pic] onto itself? (Again, be specific: what line?)

4. A regular polygon is a polygon where all sides are congruent and all angles are congruent (equilateral triangle, square, etc.).

a. How many lines of symmetry does a regular polygon with n sides have?

b. What is the degree measure of the smallest rotation that will map a regular polygon with n sides onto

itself?

READ: Orientation refers to the direction in which points go around a figure. In two-dimensional geometry, the only orientations are clockwise and counter-clockwise.

(CAT is oriented clockwise. (DOG is oriented counter-clockwise.

5. Trapezoid ABCD is reflected over line [pic]. Tell if each of the following is

preserved (stays the same) or not.

a. The length of [pic]. b. The measure of (D

c. The slope of [pic] d. The orientation of ABCD

e. The area of ABCD f. The “parallelism” of [pic] and[pic].

6. Parallelogram ABCD is rotated 90° around point A. Tell if each

of the following is preserved or not.

a. The length of diagonal [pic]. b. The measure of (B

c. The slopes of [pic] and [pic] d. The parallelism of [pic] and [pic]

e. The orientation of ABCD f. The area of ΔABC

7. In the diagram at right, [pic] is rotated clockwise around point P by an amount equal in measure to (APB.

a. What is the image of [pic]?

b. What must be true for the image of A to be B?

d. If [pic] what property of rotations guarantees that the image of A will be B?

8. In the diagram at right, [pic] is reflected over [pic].

a. What must be true for the image of [pic] to be [pic]?

b. If (APQ ( (BPQ what property of reflections guarantees that the image of [pic] will be [pic]?

c. In addition to the above, what must be true for the image of A to be B?

d. If [pic] what property of reflections guarantees that the image of A will be B?

9. In the diagram, points A and B are on line l, which is neither horizontal nor vertical. [pic] is horizontal and [pic] is vertical, making (ABC a right triangle. Line l' and triangle AB'C' are the images of line l and (ABC after a 90( rotation about A.

a. What is the measure of (BAB' ?

b. What is the relationship between lines l and l' ?

c. Give the lengths of [pic]and [pic] Justify your answers.

d. What is the slope of line l? What is the slope of line l' ?

e. How does the slope of line l' compare to the slope of line l (two words)?

10. In each of the following, describe in words a sequence of two transformations after which (DOG is the image of (CAT. Note: There are several possible correct answer to each.

a. b.

Geometry HW: Transformations - 5

Name

0. Using a compass and straight edge, construct the

image of (ABC after a dilation of scale factor 3

centered at point P.

1. In the diagram at right, O is the center of dilation and Dk(ΔOQR) = ΔOPS.

a. What is the image of R under the dilation?

b. Dk(Q) = c. Dk(O) = d. Dk[pic] =

e. If P is the midpoint of [pic], what is the constant of dilation k?

f. Using the value of k from part e, if SP = 6, find RQ.

2. a. In the diagram, [pic] is the image of [pic] after a dilation of scale factor k with center P. Which ratio is equal to the scale factor k of the dilation? (Multiple choice.)

(1) [pic] (2) [pic] (3) [pic] (4) [pic]

b. In the same diagram, suppose [pic] is the image of [pic] after a dilation of scale factor k with center P. Which ratio is equal to the scale factor k of the dilation? (Multiple choice.)

(1) [pic] (2) [pic] (3) [pic] (4) [pic]

3. In the diagram (which is NOT to scale), (PQR is the image of (ABC after a dilation of scale factor k with center P.

a. What is the scale factor of the dilation? (Think hard about this.)

b. What is the length of [pic]?

c. What is the length of [pic]?

4. In the diagram, rectangle BCDE is the image of rectangle ABFG after a dilation centered at O. OA = 4, AB = 6, and AG = 8.

a. What was the scale factor of the dilation?

b. What re the dimensions, base and height, of rectangle BCDE?

5. a. Graph ΔABC with vertices A(1, 3), B(4, 1), and

C(1, 1).

b. Graph ΔA'B'C', the image of ΔABC after a dilation of

scale factor 3 centered at the origin.

c. Find the lengths of [pic] and [pic] in simplest radical form.

d. How many times longer is [pic] than [pic]?

e. Find the areas of ΔABC and ΔA'B'C'.

f. How many times larger is the area of ΔA'B'C' than the area of ΔABC?

6. A certain hexagon has a perimeter of 30 and an area of 54.

a. Find the perimeter and the area of the hexagon after a dilation of 3.

b. Find the perimeter and the area of the hexagon after a dilation of 1/2.

7. a. Graph the line l,[pic].

b. Graph the image of l after a dilation of 3 in the

origin. What is the equation of the image?

c. Graph the line k, [pic]

d. Graph the image of k after a dilation of 3 in the

origin. What is the equation of the image?

8. If the line y = mx + b is dilated by a scale factor of k centered at the origin, in the image

(1) Both the slope and the y-intercept will be multiplied by k.

(2) The slope will be multiplied by k but the y-intercept will be unchanged.

(3) The y-intercept will be multiplied by k but the slope will be unchanged.

(4) Neither the slope nor the y-intercept will be changed.

9. Line ( is shown at right.

a. What will be the image of line ( after a dilation of scale factor 2 centered at point P?

b. Graph the image of line ( after a dilation of scale factor 2 centered at point Q. (Pick two convenient points on the line and find the image of each. Two points determine a line.)

c. How does the image of line ( in part ‘b’ compare to the preimage?

10. Line segment [pic] , whose endpoints are (6, –4) and (–3, 8), is the image of [pic] after a dilation of [pic] centered at the origin. What is the length of [pic] ?

(1) [pic] (2) 5 (2) 15 (4) 45

Geometry HW: Transformations – 6

Name

Evaluate each of the compositions in #1 – 4:

1. ry-axis o D3(3, –4) 2. RO o T2, –3 (3, –1)

3. rx-axis o ry = x (4, 2) 4. R90° o ry = 3 (–1, 2)

Use your knowledge of transformations and common sense to answer #5 - 7

5. The composition R30° o R60° is equivalent to what single transformation?

6. The composition D2 o D3 is equivalent to what single transformation?

7. The graph shows (ABC and (DEF.

a. Let (A'B'C' be the image of (ABC after a rotation about point C. State the location of B' if the location of point A ' is (–5, –3).

b. Name a second transformation that will take (A'B'C' onto (DEF.

8. Triangle QCK undergoes thecomposition [pic].

a. Aaron does the composition correctly. Graph

(Q'C'K', Aaron’s image of (QCK.

b. Name the single transformation for which the image of (QCK would be (Q'C'K'.

c. Zyta does the composition backwards. Graph

(Q"C"K", Zyta’s image of (QCK.

d. Name the single transformation for which the image of (QCK would be (Q"C"K".

e. Name the single transformation for which the image of (Q"C"K" would be (Q'C'K'.

9. (ABC is graphed at right along with three transformations of it. For each triangle, a, b and c, write a composition of transformations for which the given triangle would be the image of (ABC. Note: There is more than one right answer for each of these. Try to keep it simple.

a.

b.

c.

10. Given that TRAP and T'R'A'P' are congruent in the diagram, describe precisely a sequence of transformations that would map quadrilateral TRAP onto T'R'A'P'.

11. a. Graph the points (0, –2), (1, –1), (3, –2) and (2, –3). Connect them as shown at right: (To

the unimaginative, that may look like a lopsided arrow but it’s actually a dinosaur footprint.)

b. Draw the image of the figure from part a after the composition [pic]. Note: We are only interested in the final image. If you need an intermediate step, consider making it a different color or dashed/dotted or erasing it when you’re done.

c. Repeat part b three more times, each time starting with the previous image. (It should get easier as you go along.)

d. Is the composition [pic] commutative? In general, when will the composition of a line reflection and a translation be commutative?

Note: This composition is an example of a glide reflection. This is a term that does not seem to appear in the Common Core so you probably don’t need to worry about it.

Geometry HW: Transformations - 7

Name

1. Use compass and straight-edge to construct the image of the triangle shown after a translation along vector PQ.

2. Use compass and straight-edge to construct the image of the triangle shown after a reflection over line l.

3. Use compass and straight-edge to construct the image of the triangle shown after a counterclockwise rotation of m(A around point P.

5. The image of the point A(–2, 7) after a reflection in line k is (8, 1). Find the equation of line k.

6. In the diagram, (RST is the image of (ABC after a dilation centered at P. Which ratio does not represent the scale factor of the dilation?

(1) [pic] (3) [pic]

(2) [pic] (4) [pic]

7. a. What is the minimum rotation needed to carry a regular octagon onto itself?

b. A regular polygon has a minimum rotation of 24( to carry the polygon onto itself. How many sides does the polygon have?

8. In the diagram at right, (JKL is the image of (ABC after a dilation centered at the origin. The coordinates of the vertices are A(3, 0), B(4, 5), C(6, 0), J(5, 0), [pic] and L(10, 0). Which of the following is the ratio of the lengths of [pic] to [pic]? (Note: there is an easy way to do this and a hard way. Try to find the easy way.)

(1) [pic] (3) [pic]

(2) [pic] (4) [pic]

9a. The line 2x + 3y = 12 is dilated by a scale factor of 3 centered at the origin. What is the equation of the image of the line?

b. The same line is dilated by a scale factor of ½ from the point (9, –2). What is the equation of the image?

10. Give a precise composition of transformations what will take the letter R at right onto its image.

11. Describe precisely a sequence of transformations that will take (ABC below onto its image (XYZ.

Geometry HW: Transformations - 8

Name

1. On the grids below, triangle P is the pre-image for all transformations. Match the correct image with each of the transformations below. Images may be used more than once or not at all.

a. ry-axis b. ry = 1 c. ry = x d. R180( e. T–4, 0 f. R90(

g. rx = 1 h. T0, –10 i. R270( j. rx-axis k. R90(, (5, 5) l. R180(, (1, 5)

m. R90(, (3, 3) n. rx-axis ( ry-axis o. ry = 5 ( rx-axis

p. R90(,0 (1, 5) ( T–4, q. R90(, (1, 5) ( T–4, 0 r. ry-axis ( ry = –x

2. a. What is a rigid motion? Name the three basic ones.

b. In a rigid motion,

1) Is collinearity always preserved? 2) Is angle measure always preserved?

3) Is parallelism always preserved? 4) Are midpoints always preserved?

5) Is orientation always preserved? 6) Is slope always preserved?

3. Which of the following transformations is not a rigid motion?

(1) Line reflection (2) Rotation (3) Translation (4) Dilation

4. Which of the following transformations does not preserve orientation?

(1) Line reflection (2) Rotation (3) Translation (4) Dilation

5. Tell whether each figure represents a rigid motion.

a. b. c. d.

e. f. g. h.

6. In each diagram below, identify a specific rigid motion that would take (ABC onto its image as shown. As usual, there can be more than one right answer.

a. b. c.

d. e. f.

Geometry HW: Transformations - 9

Name

1. What is necessary for two line segments to be congruent?

2. What is necessary for two angles to be congruent?

3. In the diagram at right, ΔCDE ” ΔABF.

a. Name three pairs of congruent angles.

b. Name three pairs of congruent sides.

4. If all four pairs of corresponding angles of two quadrilaterals are congruent, must the quadrilaterals be congruent? Draw a diagram to justify your answer.

5. If all four pairs of corresponding sides of two quadrilaterals are congruent, must the quadrilaterals be congruent? Draw a diagram to justify your answer.

6. The two triangles shown at right are congruent.

a. Complete the congruence statement: (ABM (

b. Describe a rigid motion that would take the first figure onto the second.

7. The two triangles shown at right are congruent.

a. Complete the congruence statement: (JKL (

b. Describe a rigid motion that would take the first figure onto

the second.

8. The two quadrilaterals shown at right are congruent.

a. Complete the congruence statement: MATH (

b. Describe a rigid motion that would take the first figure onto the second.

9. The two quadrilaterals shown at right are congruent.

a. Complete the congruence statement: ABCD (

b. Describe a rigid motion that would take the first figure onto

the second.

10. In the diagram, ΔABC ( ΔDEF. If AB = 2x + 6, BC = 3x + 2, CA = 5x – 8 and EF = 4(x – 1), find the numerical value of the perimeter of (ABC.

11. In the diagram (not drawn to scale), ΔBIG ( ΔCAT.

Find the numerical measure of (T.

Geometry HW: Transformations - Review

Name

1. Using a compass and straight edge,

construct the image of [pic] after a

reflection over line (.

2. Using a compass and straight edge,

construct the image of [pic] after a

translation that maps A to A'.

3. Using a compass and straight edge,

construct the image of [pic] after a

rotation about point P that maps

[pic]onto [pic]

4. Pont A' is the image of point A after a

line reflection. Using a compass and

straight edge, construct the line of

reflection.

5. Segment A'B' is the image of [pic] after a rotation.

Using a compass and straight edge, locate the

center of the rotation. (Note: If a rotation carries

A onto A', then center of rotation must be on the

line of reflection that would carry A onto A'.)

6. The points A and A' have coordinates ((5, 3) and (1, 3) respectively. Under which transformations will A' be the image of A?

a. A reflection over the line y = x.

b. A 90( counterclockwise rotation about the point (3, 1).

c. A translation 4 units left.

d. A dilation of scale factor 2 centered at the point (7, 3).

e. A reflection over the line x = 3.

f. A 180( rotation about the point (3, 3).

g. A dilation of scale factor 1/3 centered at the point (–1, 3).

7. The image of point A(–4, 7) after a 180( rotation around point P is A'(8, 1). What are the coordinates of point P?

8. A translation moves P(5, –2) to P'(1, 3). What is the image of (–2, 3) under the same translation?

9. If the image of point A(–3, 5) after a reflection in line k is A'(5, 3), find the equation of line k.

10. Under the translation (x, y) ( (x + 3, y – 5), what is the pre-image of the point (1, –1)?

11. The vertices of (JKL have coordinates J(5, 1), K(–2, –3) and L(–4, 1). Under which transformation is the image (J'K'L' not congruent to (JKL?

(1) a translation of two units to the right and two units down

(2) a counterclockwise rotation of 180 degrees around the origin

(3) a reflection over the x-axis

(4) a dilation with a scale factor of 2 and centered at the origin

12. A regular polygon has a minimum rotation of 40° to carry the polygon onto itself.

a. How many sides does the polygon have?

b. What other (positive) rotations will carry this polygon onto itself?

13. In the diagram at right, (OCD is the image of (OAB after a dilation centered at the origin. The coordinates of the vertices are O(0, 0), A(0, 4), B(3, 0), C(0, 6), and D(4.5, 0). The ratio of AB to CD is

(1) [pic] (3) [pic]

(2) [pic] (4) [pic]

14. The line [pic] is transformed by a dilation centered at the point (0, 3). Which linear equation could be its image?

(1) [pic] (2) [pic] (3) [pic] (4) [pic]

15. The image of (ABC after a dilation of scale factor k centered at point A is (ADE, as shown in the diagram below.

Which statement is not always true?

(1) 2AB = AD (3) AC = CE

(2) [pic] (4) [pic]

16. Triangle GRL is the image of (BOY after a dilation of scale factor 2. If BO = x + 3 and GR = 3x – 1, then the length of GR is

(1) 5 (2) 7 (3) 10 (4) 20

17. Line segment [pic] , whose endpoints are (4, –2) and (10, –2), is rotated 90( clockwise about the origin, then dilated by a scale factor of [pic] centered at the origin, reflected over the y-axis, and finally translated 3 units right and 2 units up. What is the length of the resulting image?

(1) 9 (2) 2 (3) 6 (4) 4

18. Which of the following could not be the minimum number of degrees required to rotate a regular polygon onto itself?

( 1) 15 (2) 22.5 (3) 32 (4) 40

19. Which is the image of under the transformation rx-axis o R90° ?

(1) (2) (3) (4)

20. Line ( is mapped onto line m by a dilation centered at the origin with a scale factor of 5. The equation of line ( is 3x – 2y = 8. Write an equation for line m.

21. Triangle ABC is graphed at right. Graph (A'B'C', the image of (ABC after a 90(rotation about point P followed by a reflection over line l.

22. The point A is translated right 4 units and up 3 units; call the image B.

a. What is the slope of [pic]?

b. What is the length of[pic]?

Segment AB is dilated by a scale factor of 6 centered on point C which is not on [pic].

c. What is the slope of the image [pic]?

d. What is the length of the image [pic]?

23. In the diagram (which is NOT to scale), (PIG is the image of (COW

after a dilation of scale factor k with center S.

a. What is the scale factor of the dilation?

b. What is the length of [pic]?

c. What is the length of [pic]?

24. A square is graphed at right.

a. Write equations for all the line of symmetry for the square.

b. Describe the square’s rotational symmetries.

25. Triangle ABC has perimeter 42 and area 84; (A'B'C' is the image of (ABC after the transformation [pic].

a. What is the perimeter of (A'B'C'? b. What is the area of (A'B'C'?

26. The letter R is graphed at the origin. Describe a sequence of transformations that would take the original R into each of the three transformed Rs labeled a, b and c.

27. Given that IBEX ( TAHR, describe a sequence of transformations that will take IBEX onto TAHR.

Stuff you should know:

Properties of transformations

Orientation

Isometries

Direct and opposite

Preserved properties (length, angle measure, orientation, parallelism, etc.)

Stuff you should know:

Vocabulary:

Transformation

Reflection, line of reflection

Rotation, center of rotation, angle of rotation

Translation, vector

Dilation, center of dilation, constant of dilation

Image and pre-image

Isometry/Rigid motion

Orientation; direct and opposite isometries

Symmetry (line and rotation)

Composition

Invariant/preserved

Fixed points

Congruent

Regular polygon

Be able to:

Reflect a figure over a given line

Rotate a figure a given angle about a given point

Translate a figure along a given vector

Dilate a figure by a given scale factor from a given center

Evaluate a composition of transformations

State a transformation or composition of transformations that will take a given figure onto its image

Identify symmetries of figures

Identify fixed (invariant) points of transformations

State properties of transformations that are invariant (preserved)

Find perimeters and areas after dilations

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Remember, there are other possible correct answers.

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