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

Grade 10 Math Worksheet - Challenge Me

Circle the best answer.

1. 1.

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

   • A half-turn about (1, 1)
   • A turn of 90° about (1, 1)
   • A half-turn about the origin
   • A slide along the line y = x
2. 2.

   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?

   • Two reflections keep orientation, so they make some turn
   • Each keeps OP, and the sweeps add: 2α + 2(θ − α) = 2θ
   • The angle between the two images equals θ
   • Each reflection turns a point by θ, so the two together make θ + θ
3. 3.

   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
4. 4.

   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), the midpoint of both AC and BD; a parallelogram
   • (2, 2); ABCD is a rectangle
   • (−0.5, 0), the midpoint of AB; ABCD is a parallelogram
   • (0, 0), the origin; ABCD is a parallelogram
5. 5.

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

   • (3, 1)
   • (5, 5)
   • (0, 0)
   • (2, 2)
6. 6.

   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 half-turn about the origin
   • A turn of 90° about the origin
   • A reflection across y = x
7. 7.

   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 72°, which maps the pentagon onto itself
   • A reflection across a third line of symmetry of the pentagon
   • A turn of 144° about the center
   • A turn of 36° about the center
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 P(2, 1)
   • Slide 2 right and 1 up, then turn 90° counterclockwise about P
   • Turn 90° counterclockwise about (0.5, 1.5)
   • Turn 90° counterclockwise about the origin, then slide 2 right and 1 up
9. 9.

   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?

   • (5, −5) after a turn, or (−1, −5) after a reflection
   • (5, −5) alone, since the motion must keep the orientation
   • (2, −8), 3 past B′ in a straight line
   • (6, 2), the slide of C by (1, −2)
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
    • Right: (x, y) → (x, −y) → (y, x)
    • Wrong: a reflection and then a turn make a turn