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?

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

   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
3. 3.

   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
   • Each keeps OP, and the sweeps add: 2α + 2(θ − α) = 2θ
   • The angle between the two images equals θ
   • Two reflections keep orientation, so they make some turn
4. 4.

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

   • x = 2, through the midpoint
   • y = −x/3 + 11/3, through P and P′
   • y = 3x, through the origin
   • y = 3x − 3
5. 5.

   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
   • C is the midpoint of AA′, so it splits AA′ in half
   • AA′ passes through C, which is fixed by the turn
   • CA = CA′, since a turn keeps distance to its center
6. 6.

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

   • (2, 2)
   • (0, 0)
   • (5, 5)
   • (3, 1)
7. 7.

   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?

   • A turn of 90° about the origin
   • A reflection across y = x
   • A half-turn about the origin
   • No other motion is forced
8. 8.

   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 turn of 36° about the center
   • A turn of 144° about the center
   • A turn of 72°, which maps the pentagon onto itself
   • A reflection across a third line of symmetry of the pentagon
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: the slopes differ, so no slide or turn makes them match
   • Yes: both are 5 long; slide A to C, then turn about C
   • Yes: the slide 3 left and 2 down
   • No: just a reflection could do it, and that is not rigid
10. 10.

    Pia reflects across the x-axis and then turns 90° counterclockwise about the origin, and claims the result is one reflection across y = x. Which verdict is right?

    • Wrong: it is a reflection across y = −x
    • Wrong: it is a 90° turn about the origin
    • Wrong: a reflection and then a turn make a turn
    • Right: (x, y) → (x, −y) → (y, x)