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

Grade 10 Math Worksheet

Circle the best answer.

1. 1.

   Kira claims that reflecting across y = x and then turning 90° counterclockwise about the origin gives the same result in either order. Test it on D(3, 1). Which verdict is right?

   • Right: both orders send D to (−3, 1)
   • Wrong: a reflection then a turn is not a rigid motion
   • Wrong: D lands on (−3, 1) one way and (3, −1) the other
   • Right: both moves keep lengths, so the two results must match
2. 2.

   Triangle A(1, 1), B(4, 1), C(1, 3) must land on A′(−1, −1), B′(−4, −1), C′(−1, −3), corner to corner. Which sequence does Luz find fails?

   • Reflect across the y-axis, then across y = x
   • Reflect across y = x, then across y = −x
   • A half-turn about the origin
   • Reflect across the x-axis, then across the y-axis
3. 3.

   Noah maps quadrilateral ABCD onto PQRS by a dilation with factor 2 and then a rotation, and says the shapes are congruent. Which verdict is right?

   • Shown: the shapes have the same angles, so they are congruent
   • Not shown: a dilation with factor 2 doubles every length
   • Shown: the rotation is rigid, so the two shapes are congruent
   • Not shown: the rotation changes the angles of ABCD
4. 4.

   Tala reflects a figure across the x-axis and then across y = x. The lines meet at the origin at 45°. Which single motion does the same?

   • A turn of 45° counterclockwise about the origin
   • A reflection across the y-axis
   • A turn of 90° counterclockwise about the origin
   • A turn of 90° clockwise about the origin
5. 5.

   Ari wants two reflections in lines through the origin that together turn a figure 120° counterclockwise. Which pair works?

   • The x-axis, then the line at 60°
   • The x-axis, then the y-axis
   • The line at 60°, then the x-axis
   • The x-axis, then the line at 120°
6. 6.

   A reflection maps P(1, 5) to P′(5, 1). Which line is the mirror line, and why does Bo know?

   • y = −x + 6: the line through P and P′
   • y = x: through the midpoint (3, 3), at right angles to PP′
   • y = −x: it is at right angles to PP′
   • x = 3: the vertical line through the midpoint (3, 3) of PP′
7. 7.

   A rigid motion maps A(0, 0) to A′(2, 1) and B(3, 4) to B′. Which point does Esme find could be B′?

   • (4, 5)
   • (7, 2)
   • (6, 4)
   • (5, 4)
8. 8.

   Lines ℓ and m are perpendicular and meet at P. Jun turns both 30° about a point O, not P. What must be true of the images ℓ′ and m′?

   • They meet at 30° plus 90°, 120°
   • They are perpendicular and meet at P′, the image of P
   • They are perpendicular just when O is P
   • They are parallel to ℓ and m and still meet at the point P
9. 9.

   AB = DE, AC = DF and ∠A = ∠D. Kira slides A onto D, then turns until ray AB lies on ray DE. Why does C then land on F?

   • C lands on F once BC = EF is checked as well, as in SSS
   • C lands on F because the two triangles have equal areas
   • C lands on F, or on F's mirror image in DE; a reflection in DE then fixes it
   • C must land on F, since rigid motions keep lengths and angles
10. 10.

    Luz slides parallelogram ABCD 5 units right and then reflects it across the x-axis. Which statement about the image A′B′C′D′ is true?

    • Opposite sides stay parallel and equal, but the corners run the other way
    • Its corners run in the same order as ABCD, as a slide keeps order
    • Its sides are each 5 units longer, stretched by the slide
    • Its opposite sides may no longer be parallel after the reflection