Symmetry of polygons: lines of symmetry and rotations that map a shape onto itself

Grade 10 Math Worksheet - Challenge Me

Circle the best answer.

1. 1.

   Ari reflects a figure across x = 1, then across x = 4. What single motion does the same, and why?

   • A 180° turn about (2.5, 0), the point between the lines
   • A slide 3 units right, the distance between the lines
   • A slide 6 units right, twice the distance between the lines
   • A slide 6 units left, twice the distance between the two lines

2. Read this first

   Two mirror lines meet at the origin at a 45° angle

   The x-axis and the line y = x. A figure is reflected across the x-axis first, then across y = x.

3. 2.

   Bo reflects a figure across the x-axis, then across y = x. What single motion does the same, with its center and angle?

   • A 90° clockwise turn about (0, 0), twice the 45° between the lines
   • A 90° counterclockwise turn about (0, 0), twice the 45° between the lines
   • A 45° counterclockwise turn about (0, 0), the same as the angle between the lines
   • A reflection across the y-axis, which swaps left and right
4. 3.

   Jun reflects a figure across x = −1, then across y = 3. What single motion does the same, with its center and angle?

   • A 180° turn about (0, 0), the origin
   • A 180° turn about (−1, 3), twice the 90° between the lines
   • A 90° turn about (−1, 3), which is the same as the angle between the lines
   • A slide 2 left and 6 up, twice each coordinate
5. 4.

   Kira says the square with corners (1, 1), (5, 1), (5, 5) and (1, 5) has more symmetries than a rectangle. Which list is complete?

   • Four reflections (x = 3, y = 3, y = x, y = −x + 6) and a 180° turn about (3, 3) alone
   • Two reflections (x = 3 and y = 3) and turns of 90°, 180° and 270° about (3, 3)
   • Four reflections (x = 3, y = 3, y = x, y = −x + 6) and turns of 90°, 180° and 270° about (0, 0)
   • Four reflections (x = 3, y = 3, y = x, y = −x + 6) and turns of 90°, 180° and 270° about (3, 3)
6. 5.

   Luz draws a house shape on a grid. Which list gives all of its symmetries?

   • One reflection, across x = 2; no turn maps the house onto itself
   • No symmetry: the five sides of the house differ in length
   • Reflections across x = 2 and y = 2.5, one vertical and one horizontal
   • One reflection, across x = 2, and a 180° turn about (2, 2.5)
7. 6.

   Milo plots a right triangle with legs 4 and 4. Which list gives all of its symmetries?

   • Reflections across y = x and the x-axis, and a 90° turn about (0, 0)
   • One reflection, across y = x, and a 180° turn about (2, 2)
   • One reflection, across y = x, and no turns
   • No symmetry, because a right triangle has no fold

8. Proof

   1. Let ℓ be the bisector of ∠A. 2. The reflection across ℓ maps ray AB onto ray AC. 3. ? 4. So the reflection maps ∠B onto ∠C. 5. Reflections keep angle measure, so ∠B = ∠C.

9. 7.

   Pia's proof that the base angles of isosceles △ABC (AB = AC) are equal has a gap in step 3. Which statement belongs there?

   • Since ∠B = ∠C, the reflection must send B to C, because equal angles fix the points
   • AB = AC, so B is as far from A on the ray as C is, and the reflection puts B on C
   • BC maps onto itself, so B and C each stay fixed in place under the reflection across ℓ
   • A is the vertex, so A and C swap places under the reflection across ℓ
10. 8.

    Ren claims: every quadrilateral with a line of symmetry has a pair of equal opposite sides. Which figure breaks the claim?

    • A rectangle: it has mirror lines, and both pairs of opposite sides are equal
    • A rhombus: it has mirror lines, and the four sides are equal in length
    • A kite with sides 3, 3, 5, 5 in order: a mirror line, but no equal opposite sides
    • An isosceles trapezoid: it has a mirror line, and its two slanted sides are equal
11. 9.

    Tala claims: every figure with a half-turn symmetry also has a line of symmetry. Which figure breaks the claim?

    • A rectangle: it has a half turn, and it also has two lines of symmetry through its center
    • A regular hexagon: it has a half turn, and it also has mirror lines
    • A slanted parallelogram with unequal sides: it has a half turn, but no mirror line
    • A rhombus: it has a half turn, and it also has mirror lines
12. 10.

    Zee knows a quadrilateral maps onto itself by a half turn about the crossing of its diagonals. Which property can this symmetry NOT prove?

    • Opposite sides are equal
    • The diagonals bisect each other
    • The diagonals are perpendicular
    • Opposite angles are equal