Reflect over any line and rotate about a point, with coordinate rules

Grade 10 Math Worksheet - Challenge Me

Circle the best answer.

1. 1.

   A rotation maps A(3, 1) to A′(1, 3) and B(4, 2) to B′(0, 4). Where is the center, and why does Milo know?

   • (2, 3): the midpoint of BB′
   • (2, 2): the midpoint of AA′
   • (1, 1): on the perpendicular bisectors of AA′ and BB′
   • (0, 0): rotations on this grid turn about the origin
2. 2.

   Noah reflects a figure across y = x and then across x + y = 2. Which single motion does the same?

   • A half-turn about the origin
   • A turn of 90° about (1, 1)
   • A half-turn about (1, 1)
   • A slide along the line y = x
3. 3.

   Pia reflects a figure across y = x and then across y = x + 4. These lines are parallel. Which single slide does the same?

   • (−4, 4): twice the 2√2 gap, across the lines
   • (4, −4): twice the gap, but back toward the first line
   • (−2, 2): the gap between the lines
   • (0, 8): twice the 4-unit vertical gap
4. 4.

   Lines ℓ and m meet at O at angle θ. Ren explains why reflecting across ℓ, then m, turns every point 2θ about O. Which reason is right?

   • Each reflection turns a point by θ, so the two together make θ + θ
   • Two reflections keep orientation, so they make some turn
   • The angle between the two images equals θ
   • Each keeps OP, and the sweeps add: 2α + 2(θ − α) = 2θ
5. 5.

   A figure maps onto itself under reflection across the x-axis and under reflection across the y-axis. Which other motion must Esme find maps it onto itself?

   • No other motion is forced
   • A turn of 90° about the origin
   • A half-turn about the origin
   • A reflection across y = x
6. 6.

   Two lines of symmetry of a regular pentagon meet at its center at 36°. Jun reflects across one and then the other. What single motion results?

   • A reflection across a third line of symmetry of the pentagon
   • A turn of 72°, which maps the pentagon onto itself
   • A turn of 36° about the center
   • A turn of 144° about the center
7. 7.

   Kira's parallelogram has unequal sides and no right angles. Which motions, other than doing nothing, map it onto itself?

   • A reflection across the line through the midpoints of opposite sides
   • A reflection across either diagonal
   • A turn of 90° about the center
   • Only the half-turn about the point where the diagonals cross
8. 8.

   “each square one unit” Triangle A(0, 0), B(4, 0), C(0, 3) must land on P(2, 1), Q(2, 5), R(−1, 1), corner to corner. Which sequence does Luz find fails?

   • Turn 90° counterclockwise about the origin, then slide 2 right and 1 up
   • Turn 90° counterclockwise about (0.5, 1.5)
   • Turn 90° counterclockwise about P(2, 1)
   • Slide 2 right and 1 up, then turn 90° counterclockwise about P
9. 9.

   Segment AB runs from (1, 2) to (4, 6). Segment CD runs from (−2, 0) to (2, −3). Is there a rigid motion taking A to C and B to D, Milo asks?

   • No: just a reflection could do it, and that is not rigid
   • Yes: the slide 3 left and 2 down
   • Yes: both are 5 long; slide A to C, then turn about C
   • No: the slopes differ, so no slide or turn makes them match
10. 10.

    Coordinate grid, all four quadrants, each square one unit, origin in the middle. A rigid motion maps A(1, 1), B(5, 1), C(5, 4) to A′(2, −1), B′(2, −5) and C′. Which statement about C′ does Noah find is right?

    • (2, −8), 3 past B′ in a straight line
    • (6, 2), the slide of C by (1, −2)
    • (5, −5) alone, since the motion must keep the orientation
    • (5, −5) after a turn, or (−1, −5) after a reflection