the regular octagon has a perimeter of 122.4 cm. which statements about the octagon are true? select two…

Explanation:

Step1: Find side - length of octagon

A regular octagon has 8 equal sides. Given perimeter $P = 122.4$ cm. Let the side - length be $s$. Then $P=8s$. So, $s=\frac{122.4}{8}=15.3$ cm. Segment $YZ$ is a side of the octagon, so the length of segment $YZ$ is $15.3$ cm.

Step2: Analyze central angle

The measure of the central angle of a regular $n$ - gon is given by $\theta=\frac{360^{\circ}}{n}$. For an octagon ($n = 8$), $\theta=\frac{360^{\circ}}{8}=45^{\circ}$. $\angle ZXW$ is a central angle, so $\angle ZXW = 45^{\circ}$.

Step3: Analyze radius and apothem angle

The angle between the radius and the apothem of a regular octagon is $\frac{45^{\circ}}{2}=22.5^{\circ}$, not $30^{\circ}$.

Step4: Analyze length of $XY$

The radius $r = 20$ cm, half of the side - length $s/2=7.65$ cm. By the Pythagorean theorem, if the apothem is $a$, then $a=\sqrt{20^{2}-7.65^{2}}$, so the length of segment $XY$ (apothem) can be found by solving for $a$ in $20^{2}-7.65^{2}=a^{2}$.

Step5: Analyze length of $WZ$

$WZ$ is not a radius. The radius is $20$ cm, and $WZ$ is not equal to the radius.

Answer:

The length of segment $YZ$ is $15.3$ cm; The measure of the central angle, $\angle ZXW$, is $45^{\circ}$; The length of segment $XY$ can be found by solving for $a$ in $20^{2}-7.65^{2}=a^{2}$.