Question

finding missing measures in regular polygons
a regular pentagon has a perimeter of 60 cm.
what is the measure of angle cbd?
$\boldsymbol{square}^circ$
what is the length of segment cd?
$\boldsymbol{square}$ cm
which trigonometric ratio can be used to compare bc to cd using angle cbd?
$\boldsymbol{square}$
what is the approximate length of bc?
$\boldsymbol{square}$ cm
what is the approximate length of bd?
$\boldsymbol{square}$ cm

Explanation:

Step1: Find side length of pentagon

A regular pentagon has 5 equal sides. Given perimeter $P=60$ cm, side length $s=\frac{P}{5}=\frac{60}{5}=12$ cm. So $CD=12$ cm.

Step2: Calculate $\angle CBD$

First, find internal angle of regular pentagon: $\text{Internal angle} = \frac{(5-2)\times180^\circ}{5}=108^\circ$. Point $B$ is the center, so $\angle CBD = \frac{1}{2}\times\frac{360^\circ}{5}=36^\circ$.

Step3: Identify trigonometric ratio

In right triangle $BCD$, $\angle BCD=90^\circ$, $\angle CBD=36^\circ$. $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$, so $\tan(\angle CBD)=\frac{CD}{BC}$.

Step4: Calculate length of $BC$

From $\tan(36^\circ)=\frac{CD}{BC}$, rearrange to $BC=\frac{CD}{\tan(36^\circ)}$. Substitute $CD=12$ cm, $\tan(36^\circ)\approx0.7265$.
$BC=\frac{12}{0.7265}\approx16.52$ cm.

Step5: Calculate length of $BD$

Use Pythagorean theorem: $BD=\sqrt{BC^2+CD^2}$. Substitute $BC\approx16.52$ cm, $CD=12$ cm.
$BD=\sqrt{16.52^2+12^2}=\sqrt{272.91+144}=\sqrt{416.91}\approx20.42$ cm.

Answer:

1. Measure of $\angle CBD$: $36^\circ$
2. Length of segment $CD$: $12$ cm
3. Trigonometric ratio: $\tan(\angle CBD)=\frac{CD}{BC}$ (tangent)
4. Approximate length of $BC$: $16.5$ cm
5. Approximate length of $BD$: $20.4$ cm