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

Grade 10 Math Worksheet

Circle the best answer.

1. 1.

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

   • Reflections across y = 0 and x = 4, and a 90° 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, with no turn
   • Reflections across y = 0 and x = 4, and a 180° turn about (4, 0)
2. 2.

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

   • Reflections across x = 3.5 and y = 1.5, and a 180° turn about (3.5, 1.5)
   • No symmetry of any kind
   • No reflections; one 180° turn about (0, 0)
   • No reflections; one 180° turn about (3.5, 1.5)
3. 3.

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

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

   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?

   • 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°
   • AB = AD = √17, so the two sides at A are equal and the kite is symmetric
5. 5.

   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?

   • 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)
   • The diagonals are perpendicular, so the corners swap in pairs
   • AB = CD = √26, so opposite sides match under the turn, and that is enough
6. 6.

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

   • The triangle has a right angle at C, so its legs are mirror images
   • 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
   • C(3, 4) is above the base, so the picture looks balanced and symmetric
7. 7.

   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
8. 8.

   In isosceles △ABC (AB = AC), the reflection across ℓ swaps B and C. Ren asks why ℓ is perpendicular to BC and crosses its midpoint. What is the reason?

   • A reflection that swaps B and C has the perpendicular bisector of BC as mirror line
   • BC is parallel to ℓ, so ℓ splits BC in half at the same level as the base
   • An angle bisector is perpendicular to the opposite side, in any triangle you pick
   • ℓ passes through A, so it must pass through the midpoint of BC as well
9. 9.

   Tala says an equilateral triangle maps onto itself by a 120° turn about its center, sending A to B to C. Why must each angle be 60°?

   • The turn sends ∠A to ∠B to ∠C, so the three angles are equal, and add to 180°
   • 120° is the turn, so each angle is 120° and the three angles add to 360°
   • A reflection keeps A fixed, so ∠A = 60° and the other two angles follow
   • The three sides are equal, so each angle is 60° by the definition of an equilateral triangle
10. 10.

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

    • A half turn about the center maps AC onto itself, so AC must equal BD
    • The reflection swaps A with B and C with D, so AC maps onto BD, and length is kept
    • 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