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 slide 6 units right, twice the distance between the lines
   • A 180° turn about (2.5, 0), the point between the lines
   • A slide 6 units left, twice the distance between the two lines
   • A slide 3 units right, the distance between the 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 reflection across the y-axis, which swaps left and right
   • A 90° clockwise 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 90° counterclockwise turn about (0, 0), twice the 45° between the lines
4. 3.

   A figure is reflected across y = 2 first, then across y = x + 2. Esme reflects a figure across y = 2, then across y = x + 2. What single motion does the same, with its center and angle?

   • A 90° counterclockwise turn about (0, 2), twice the 45° between the lines
   • A 90° clockwise turn about (0, 2), twice the 45° between the lines
   • A 90° counterclockwise turn about (0, 0), the origin
   • A 45° counterclockwise turn about (0, 2), the same as the angle between the lines
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 turns of 90°, 180° and 270° about (0, 0)
   • 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 (3, 3)
   • Four reflections (x = 3, y = 3, y = x, y = −x + 6) and a 180° turn about (3, 3) alone
6. 5.

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

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

   Noah draws an equilateral triangle with corners (0, 0), (6, 0) and (3, 3√3). Which list gives all of its symmetries?

   • Three reflections, each through a corner and the opposite midpoint, and turns of 120° and 240° about (3, √3)
   • One reflection, across x = 3, and no turns, since just the line through the top corner folds it
   • Three reflections, each through one corner and the opposite midpoint, and turns of 120° and 240° about (3, 0)
   • Three reflections, each through one corner and the opposite midpoint, and a 180° turn about (3, √3)

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?

   • BC maps onto itself, so B and C each stay fixed in place under the reflection across ℓ
   • 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
   • 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?

    • An isosceles trapezoid: it has a mirror line, and its two slanted 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
    • A rectangle: it has mirror lines, and both pairs of opposite 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 rhombus: it has a half turn, and it also has mirror lines
    • A regular hexagon: it has a half turn, and it also has mirror lines
    • A rectangle: it has a half turn, and it also has two lines of symmetry through its center
    • A slanted parallelogram with unequal sides: it has a half turn, but no mirror line
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?

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