6. How many lines of symmetry do these shapes have?

Question

Homi · Accepted Answer

Understand the Question

The Ashoka Chakra contains 24 equally spaced spokes, which form 12 pairs of opposite spokes aligned through the center.
A line of symmetry is a straight line dividing the shape into two identical mirror halves. For the chakra, each line passes through a pair of opposite spokes.
A shape possesses rotational symmetry if it matches its original appearance when rotated by an angle less than or equal to $360^\circ$ .

Step 1 · Find the Number of Lines of Symmetry

The Ashoka Chakra has $24$ spokes that form $12$ pairs of diametrically opposite spokes. Diagram 1

A line of symmetry passes through each pair of opposite spokes, dividing the wheel into identical mirror halves.

$\text{Number of lines of symmetry} = 12$

Step 2 · Find the Angles of Rotational Symmetry

Since there are $12$ pairs of symmetric spokes, the wheel aligns with itself every time it is rotated by one unit of these symmetric pairs.

\begin{aligned} \text{Smallest angle of symmetry} &= \dfrac{360^\circ}{12} \\[0.6em] &= 30^\circ \end{aligned}

The other angles of rotational symmetry are multiples of $30^\circ$ : $30^\circ, 60^\circ, 90^\circ, 120^\circ, 150^\circ, 180^\circ, 210^\circ, 240^\circ, 270^\circ, 300^\circ, 330^\circ, 360^\circ$

Answer

The Ashoka Chakra has $12$ lines of symmetry.

Smallest angle of rotational symmetry: $30^\circ$
Other angles of rotational symmetry: $60^\circ, 90^\circ, 120^\circ, 150^\circ, 180^\circ, 210^\circ, 240^\circ, 270^\circ, 300^\circ, 330^\circ, 360^\circ$

Common Mistakes

Spokes vs. Lines of Symmetry: Assuming there are $24$ lines of symmetry because there are $24$ spokes. Since each line passes through $2$ opposite spokes, there are only $\dfrac{24}{2} = 12$ lines of symmetry.
Angle of Rotation: Dividing $360^\circ$ by $24$ instead of $12$ when finding the rotational symmetry for the pairs of opposite spokes.

Question 12

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