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 slide 6 units left, twice the distance between the two lines
   • A slide 3 units right, the distance between the lines
   • A 180° turn about (2.5, 0), the point 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 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
   • A 90° clockwise 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° clockwise turn about (0, 2), twice the 45° between the lines
   • A 90° counterclockwise 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 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 (3, 3)
   • Four reflections (x = 3, y = 3, y = x, y = −x + 6) and turns of 90°, 180° and 270° about (0, 0)
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
   • One reflection, across x = 2, and a 180° turn about (2, 2.5)
   • Reflections across x = 2 and y = 2.5, one vertical and one horizontal
   • No symmetry: the five sides of the house differ in length
7. 6.

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

   • One reflection, across y = x, and a 180° turn about (2, 2)
   • No symmetry, because a right triangle has no fold
   • Reflections across y = x and the x-axis, and a 90° turn about (0, 0)
   • One reflection, across y = x, and no turns
8. 7.

   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)
   • Three reflections, each through one corner and the opposite midpoint, and turns of 120° and 240° about (3, 0)
   • 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 a 180° turn about (3, √3)

9. 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.

10. 8.

    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
    • A is the vertex, so A and C swap places under the reflection across ℓ
    • BC maps onto itself, so B and C each stay fixed in place under the reflection across ℓ
    • AB = AC, so B is as far from A on the ray as C is, and the reflection puts B on C
11. 9.

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

    • A slanted parallelogram with unequal sides: it has a half turn, but no mirror line
    • A regular hexagon: it has a half turn, and it also has mirror lines
    • A rhombus: 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
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 are perpendicular
    • Opposite sides are equal
    • The diagonals bisect each other
    • Opposite angles are equal