Question

use these three geometric figures for problems 9 to 12. (a) an equilateral triangle (3 equal side lengths) (b) a square (4 equal side lengths) (c) a regular pentagon (5 equal side lengths) 9. what is the least number of degrees that you could rotate figure (a) around its center so that it appears to be unchanged? 10. what is the least number of degrees that you could rotate figure (b) around its center so that it appears to be unchanged?

Explanation:

Step1: Recall rotational - symmetry formula

For a regular polygon with \(n\) sides, the least angle of rotation \(\theta\) about its center for which it looks unchanged is given by \(\theta=\frac{360^{\circ}}{n}\).

Step2: Solve for equilateral triangle (Figure a)

An equilateral triangle has \(n = 3\) sides. Using the formula \(\theta=\frac{360^{\circ}}{n}\), we substitute \(n = 3\): \(\theta=\frac{360^{\circ}}{3}=120^{\circ}\).

Step3: Solve for square (Figure b)

A square has \(n = 4\) sides. Using the formula \(\theta=\frac{360^{\circ}}{n}\), we substitute \(n = 4\): \(\theta=\frac{360^{\circ}}{4}=90^{\circ}\).

Answer:

1. \(120^{\circ}\)
2. \(90^{\circ}\)