Question

transformations test (topic 4)
given a regular pentagon (five - sided shape), which of the following angles of rotation about the center would not carry the figure onto itself?
288 degrees
144 degrees
72 degrees
224 degrees

Explanation:

Step1: Find the rotational symmetry angle

For a regular \( n \)-sided polygon, the smallest angle of rotational symmetry is \( \frac{360^\circ}{n} \). For a pentagon, \( n = 5 \), so the smallest rotational symmetry angle is \( \frac{360^\circ}{5}=72^\circ \). Any multiple of \( 72^\circ \) will map the pentagon onto itself.

Step2: Check each option

• For \( 288^\circ \): \( 288\div72 = 4 \), which is an integer, so \( 288^\circ \) is a multiple of \( 72^\circ \).
• For \( 144^\circ \): \( 144\div72 = 2 \), which is an integer, so \( 144^\circ \) is a multiple of \( 72^\circ \).
• For \( 72^\circ \): \( 72\div72 = 1 \), which is an integer, so \( 72^\circ \) is a multiple of \( 72^\circ \).
• For \( 224^\circ \): \( 224\div72=\frac{28}{9}\approx 3.11 \), which is not an integer, so \( 224^\circ \) is not a multiple of \( 72^\circ \).

Answer:

224 degrees (the option with 224 degrees)