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

Grade 10 Math Worksheet

Circle the best answer.

1. 1.

   Ari puts a rug on a grid. Which list gives every symmetry of the rectangle, with centers and mirror lines?

   • Reflections across x = 3, y = 2 and both diagonals, and a 180° turn about (3, 2)
   • Reflections across x = 3 and y = 2, and a 180° turn about (3, 2)
   • Reflections across x = 3 and y = 2, and a 180° turn about (0, 0)
   • Reflections across x = 3 and y = 2, with no turn
2. 2.

   Bo draws a rhombus on a grid. Which list gives every symmetry, with centers and mirror lines?

   • Reflections across y = 0 and x = 4, with no turn
   • Reflections across y = 0 and x = 4, and a 180° turn about (4, 0)
   • Reflections across the lines through midpoints of opposite sides, and a 180° turn about (4, 0)
   • Reflections across y = 0 and x = 4, and a 90° turn about (4, 0)
3. 3.

   Esme draws a slanted parallelogram on a grid. Which list gives every symmetry, with centers and mirror lines?

   • No reflections; one 180° turn about (0, 0)
   • No symmetry of any kind
   • Reflections across x = 3.5 and y = 1.5, and a 180° turn about (3.5, 1.5)
   • No reflections; one 180° turn about (3.5, 1.5)
4. 4.

   Jun draws an isosceles trapezoid on a grid. Which list gives every symmetry, with centers and mirror lines?

   • One reflection, across x = 4, and a 180° turn about (4, 1.5)
   • One reflection, across x = 4, and no turn symmetry
   • No symmetry of any kind
   • Reflections across x = 4 and y = 1.5
5. 5.

   AB = AD = √17. BD has slope -1. Kira thinks the kite ABCD is symmetric across the line y = x. Which computed fact is enough to show it?

   • AB = AD = √17, so the two sides at A are equal and the kite is symmetric
   • A and C lie on y = x, so they stay put and the other corners must match too
   • BD is perpendicular to AC, so y = x must be a mirror line for the kite
   • BD has slope −1 with its midpoint (2.5, 2.5) on y = x, so y = x bisects BD at 90°
6. 6.

   AB = CD = √26. AC = √65 and BD = √13. Luz thinks a half turn maps the quadrilateral ABCD onto itself. Which computed fact is enough to show it?

   • AB = CD = √26, so opposite sides match under the turn, and that is enough
   • The diagonals are perpendicular, so the corners swap in pairs
   • AC and BD have equal lengths, so the turn swaps the two diagonals
   • The midpoint of AC and the midpoint of BD are the same point, (3.5, 2)
7. 7.

   Milo thinks the triangle with corners (0, 0), (6, 0) and (3, 4) is symmetric across x = 3. Which computed fact is enough?

   • C(3, 4) is above the base, so the picture looks balanced and symmetric
   • The base has length 6, so each half of the base has length 3
   • x = 3 is the perpendicular bisector of AB, and C(3, 4) lies on x = 3
   • The triangle has a right angle at C, so its legs are mirror images
8. 8.

   Noah wonders whether the triangle with corners (0, 0), (6, 0) and (2, 5) has any line of symmetry. What is the best answer?

   • No: the triangle has a vertex at (2, 5), which is not on the base
   • Yes: every triangle has a line of symmetry through its longest side
   • No: the sides are all different, and a reflection must swap two equal sides
   • Yes: x = 3 is the perpendicular bisector of the base AB, so it is a line of symmetry
9. 9.

   Pia proves the base angles of isosceles △ABC (AB = AC) are equal with a reflection across the bisector of ∠A. Which reason is best?

   • AB = AC, so ∠B = ∠C by the definition of an isosceles triangle with equal sides
   • The reflection keeps B fixed, so ∠B stays the same and ∠C is unchanged too
   • The reflection swaps B and C, so it maps ∠B onto ∠C, and reflections keep angles
   • Base angles are equal in any triangle, because the base is one of its sides
10. 10.

    Zee proves the diagonals of a rectangle ABCD are equal, using the reflection across its vertical midline. Which reason is best?

    • The diagonals of any quadrilateral are equal, so these two diagonals are equal
    • The diagonals bisect each other, so they must be equal in length, as in a rectangle
    • The reflection swaps A with B and C with D, so AC maps onto BD, and length is kept
    • A half turn about the center maps AC onto itself, so AC must equal BD