Question

the regular octagon has a perimeter of 122 4 cm. which statements about the octagon are true? select two options. the length of segment yz is 15 3 cm. the measure of the angle formed by the radius and the apothem is 30°. the length of segment xy can be found by solving for a in 20² - 7 65² = a². the length of segment wz is 20 cm. the measure of the central angle ∠zxw is 45°.

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 $YZ = 15.3$ cm.

Step2: Calculate central angle

The measure of the central angle of a regular octagon is $\theta=\frac{360^{\circ}}{n}$, where $n = 8$. So $\theta=\frac{360^{\circ}}{8}=45^{\circ}$. $\angle ZXW$ is a central angle, so $\angle ZXW = 45^{\circ}$.

Step3: Analyze radius and apothem angle

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

Step4: Analyze segment lengths

$XZ$ is the radius $r = 20$ cm, $YZ = 15.3$ cm. In right - triangle $XYZ$, by the Pythagorean theorem, $XY=\sqrt{XZ^{2}-YZ^{2}}=\sqrt{20^{2}-15.3^{2}}$. And $WZ$ is not equal to the radius ($WZ$ is a chord and not a radius).

Answer:

The length of segment $YZ$ is $15.3$ cm; The measure of the central angle $\angle ZXW$ is $45^{\circ}$