Question

for each figure below, determine if it has rotational symmetry. if it does, give the smallest angle of rotation needed for the figure to appear unmoved. rotational symmetry? ○ yes ○ no angle? □° rotational symmetry? ○ yes ○ no angle? □°

Explanation:

Left Figure (Star - like with 5 arms)

Step1: Count the number of identical parts

The left figure has 5 identical arms. For a figure with \( n \) identical rotational parts, the smallest angle of rotation for rotational symmetry is \( \frac{360^\circ}{n} \). Here, \( n = 5 \).

Step2: Calculate the angle

Using the formula \( \theta=\frac{360^\circ}{n} \), substitute \( n = 5 \). So, \( \theta=\frac{360^\circ}{5}=72^\circ \). And since it can be rotated by \( 72^\circ \) to match its original position, it has rotational symmetry.

Step1: Count the number of sides (identical parts)

A heptagon has 7 identical sides (or rotational parts). The formula for the smallest rotational symmetry angle is \( \frac{360^\circ}{n} \), where \( n \) is the number of sides (or identical rotational parts). Here, \( n = 7 \).

Step2: Calculate the angle

Using \( \theta=\frac{360^\circ}{n} \), substitute \( n = 7 \). So, \( \theta=\frac{360^\circ}{7}\approx51.43^\circ \) (but we use the exact fraction or decimal as needed, and since a regular heptagon has rotational symmetry, we calculate the angle as \( \frac{360}{7} \) or its decimal approximation. However, for a regular polygon, the smallest rotational angle is \( \frac{360^\circ}{n} \), so for \( n = 7 \), it is \( \frac{360}{7}\approx51.43^\circ \), but we can also present it as \( \frac{360}{7}^\circ \) or the decimal. But in the context of a regular polygon, we use \( \frac{360}{n} \). So for \( n = 7 \), the angle is \( \frac{360^\circ}{7}\approx51.4^\circ \) (or exact fraction). And since a regular heptagon has rotational symmetry, the answer for rotational symmetry is yes.

Answer:

Rotational symmetry? Yes
Angle? \( 72^\circ \)

Right Figure (Heptagon - 7 - sided polygon)