Sequences of transformations, including glide reflections

Answer Key - Grade 10 Math Worksheet

Circle the best answer, or write your answer on the line.

1. Read this first

   Use this

   Coordinate grid, all four quadrants, 20 squares across. △PQR: P(1, 1), Q(4, 1), R(1, 3). Its image: P′(1, 0), Q′(1, 3), R′(-1, 0). Both are drawn.

2. 1.

   Bo moved a triangle on a grid with two moves in a row. Which two moves, in order, carry it onto its image?

   • A slide 2 right and 1 down, then a 90° counterclockwise turn about the origin
   • A 90° counterclockwise turn about the origin, then a slide 1 down
   • A 90° counterclockwise turn about the origin, then a slide 2 right and 1 down
   • A 90° clockwise turn about the origin, then a slide 2 right and 1 down
3. 2.

   Coordinate grid, all four quadrants, 20 squares across. Jun moved a triangle on a grid with two moves in a row. Which two moves, in order, carry it onto its image?

   • A 180° turn about the origin, then a slide 1 left and 6 up
   • A 180° turn about the origin, then a slide 5 left and 8 up
   • A slide 1 left and 6 up, then a 180° turn about the origin
   • A 90° counterclockwise turn about the origin, then a slide 1 left and 6 up
4. 3.

   Luz moved a triangle on a grid with two moves in a row. Which two moves, in order, carry it onto its image?

   • A flip across the line y = x, then a slide 2 left and 3 down
   • A slide 2 left and 3 down, then a flip across the line y = x
   • A flip across the line y = -x, then a slide 2 left and 3 down
   • A flip across the line y = x, then a slide 1 left and 4 down
5. 4.

   Coordinate grid, all four quadrants, 20 squares across. Milo moved a triangle on a grid with two moves in a row. Which two moves, in order, carry it onto its image?

   • A slide 3 right and 2 down, then a flip across the line y = 1
   • A flip across the line y = 1, then a slide 3 right and 2 down
   • A flip across the line y = -1, then a slide 3 right and 2 down
   • A flip across the line y = 1, then a slide 3 right and 2 up
6. 5.

   Zee's tile pattern uses a glide reflection: a slide along the mirror line y = k, then a flip across it. Corner P(-2, 5) lands on P′(7, -1). What is k?

   2
7. 6.

   Tala tests whether order matters. Image 1: flip a design across the y-axis, then slide it 6 units right. Image 2: slide first, then flip. How do the images compare?

   • Different: image 1 is 12 units to the right of image 2
   • Different: image 1 is 12 units to the left of image 2
   • The same image, because a slide and a flip are both rigid motions
   • Different: image 1 is 6 units to the right of image 2

8. Use this

   A design with corner (x, y). Image 1 is made by the half turn, then the flip. Image 2 is made by the flip, then the half turn.

9. 7.

   Esme asks whether order matters for a 180° turn about the origin and a flip across the y-axis. Image 1: turn, then flip. Image 2: flip, then turn. How do the images compare?

   • Different: order matters when one of the moves is a flip
   • The same: both orders send (x, y) to (x, -y)
   • Different: the two images are mirror images across the x-axis
   • The same: both orders send (x, y) to (-x, -y)
10. 8.

    Ren wants A to land on D, B on E and C on F, using only turns and slides. Which statement is right?

    • It cannot be done: △DEF is bigger than △ABC
    • It cannot be done: the triangles are congruent, but a flip is needed to reverse the turn direction
    • It can be done: slide A onto D, then turn 90° counterclockwise about D
    • It can be done: the three side lengths match, and turns and slides keep lengths

11. Use this

    AB = DE = 5 cm. ∠B = ∠E = 40°. BC = EF = 7 cm. A on D uses AB = DE.

12. 9.

    Pia moves △ABC so B lands on E and A lands on D, flipping if needed so C is on the same side of line DE as F. Which fact puts C exactly on F?

    • ∠B = ∠E alone: equal angles put C exactly on F
    • BC = EF alone: equal lengths put C exactly on F
    • ∠B = ∠E puts C on ray EF, and BC = EF stops C at F
    • AB = DE again: it already put A on D, so it also puts C on F
13. 10.

    Kira moves △ABC so A lands on D and B on E, with C on the same side of DE as F. Which facts put C exactly on F?

    • Both angle pairs: C is on ray DF and on ray EF, and those rays meet only at F
    • ∠C = ∠F: the third angles are equal, so C lands on F
    • AB = DE again: it already put A on D and B on E, so C follows
    • ∠A = ∠D alone: C is on ray DF, which is enough to fix C at F