Problem 10

Explanation

We analyze the lines of symmetry (reflectional symmetry) and rotational symmetry order for each given plane figure:

1. Regular Hexagon

   • Lines of symmetry: $6$ (lines through vertices and midpoints of sides).
   • Rotational symmetry order: $6$ (angles of rotation $60^\circ, 120^\circ, 180^\circ, 240^\circ, 300^\circ, 360^\circ$).
   • Conclusion: $6 = 6$. Figure $(1)$ satisfies the condition.

2. Isosceles Right Triangle

   • Lines of symmetry: $1$ (altitude from the right angle to the hypotenuse).
   • Rotational symmetry order: $1$ (only the initial position $360^\circ$).
   • Conclusion: $1 = 1$. Figure $(2)$ satisfies the condition.

3. Right Trapezoid

   • Lines of symmetry: $0$ (no axis of symmetry).
   • Rotational symmetry order: $1$ (only the initial position $360^\circ$).
   • Conclusion: $0 \neq 1$. Figure $(3)$ does not satisfy the condition.

4. Non-square Rectangle

   • Lines of symmetry: $2$ (vertical and horizontal lines through the center).
   • Rotational symmetry order: $2$ (angles of rotation $180^\circ$ and $360^\circ$).
   • Conclusion: $2 = 2$. Figure $(4)$ satisfies the condition.

Thus, the figures that have the same number of lines of symmetry as the order of rotational symmetry are figures $(1)$, $(2)$, and $(4)$. There are $3$ figures.