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′
   • (0, 0): rotations on this grid turn about the origin
   • (1, 1): on the perpendicular bisectors of AA′ and BB′
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 slide along the line y = x
   • A turn of 90° about (1, 1)
   • A half-turn about (1, 1)
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 gap, but back toward the first line
   • (−2, 2): the gap between the lines
   • (−4, 4): twice the 2√2 gap, across 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?

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

   A reflection maps P(−1, 4) to P′(5, 2). Which line does Tala find is the mirror line?

   • y = −x/3 + 11/3, through P and P′
   • x = 2, through the midpoint
   • y = 3x − 3
   • y = 3x, through the origin
6. 6.

   A rotation about C maps A to A′. Zee says C must lie on the perpendicular bisector of AA′. Which reason is right?

   • The rotation turns AA′ by 90° about its own midpoint
   • CA = CA′, since a turn keeps distance to its center
   • AA′ passes through C, which is fixed by the turn
   • C is the midpoint of AA′, so it splits AA′ in half
7. 7.

   A half-turn maps A(−2, 1) to C(6, 3) and B(1, −1) to D(3, 5). Where is its center, and what does Ari conclude about ABCD?

   • (2, 2); ABCD is a rectangle
   • (0, 0), the origin; ABCD is a parallelogram
   • (2, 2), the midpoint of both AC and BD; a parallelogram
   • (−0.5, 0), the midpoint of AB; ABCD is a parallelogram
8. 8.

   A rotation maps A(0, 4) to A′(4, 0). Which point does Bo show could NOT be its center?

   • (5, 5)
   • (2, 2)
   • (3, 1)
   • (0, 0)
9. 9.

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

   • Only the half-turn about the point where the diagonals cross
   • A reflection across either diagonal
   • A turn of 90° about the center
   • A reflection across the line through the midpoints of opposite sides
10. 10.

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