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?

   • 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 orders send D to (−3, 1)
   • 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 y = x, then across y = −x
   • A half-turn about the origin
   • Reflect across the x-axis, then across the y-axis
   • Reflect across the y-axis, then across y = x
3. 3.

   Triangle ABC reads A, B, C clockwise. Its congruent image reads A′, B′, C′ counterclockwise. Which kind of sequence could Milo use to map ABC onto A′B′C′?

   • A slide alone
   • One reflection, perhaps followed by a slide
   • Two reflections, one after the other
   • A rotation about one of the corners of triangle ABC
4. 4.

   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
5. 5.

   Pia turns a figure 90° counterclockwise about the origin and then reflects it across the x-axis. Which single motion does the same?

   • A turn of 90° clockwise about the origin
   • A reflection across y = −x
   • A half-turn about the origin
   • A reflection across y = x
6. 6.

   Ren reflects any point (x, y) across x = 1 and then across x = 4. Which line of algebra shows the result is one slide?

   • x → 2 − x → 8 + x, a slide of 8
   • x → 2 − x → 8 − (2 − x) = x + 6, a slide 6 right
   • x → 2 − x → 2 − (8 − x) = x − 6, a slide 6 left
   • x → 2 − x → 8 − x, a slide of 3, the gap
7. 7.

   Zee wants two reflections across vertical lines that together slide every point 8 units right. Which pair works?

   • y = 1, then y = 5
   • x = 1, then x = 9
   • x = 1, then x = 5
   • x = 5, then x = 1
8. 8.

   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′?

   • (5, 4)
   • (6, 4)
   • (7, 2)
   • (4, 5)
9. 9.

   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
10. 10.

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