Respuesta :
The options that are true are;
2. The measure of the angle formed by the radius and the apothem is 30°.
4. In a regular hexagon, the radius and the side length are equal in length.
5. The area of the hexagon is about 221.7 square cm.
Methods used for the required calculations
The shape of the given polygon = Hexagon
The length of the apothem, a = 8 cm
The side length of a regular polygon is given by the following formula;
- [tex]s = \mathbf{a \times 2 \times tan \left(\dfrac{180^{\circ}}{n} \right)}[/tex]
Where;
n = The number of sides of the polygon
The side length of the given polygon is therefore given as follows;
[tex]s = 8 \times 2 \times tan\left(\dfrac{180^{\circ}}{6} \right) = 16 \times \dfrac{1}{\sqrt{3} } \approx \mathbf{9.24}[/tex]
The perimeter of the hexagon is therefore;
[tex]P = 6 \times s = 6 \times 16 \times \dfrac{1}{\sqrt{3} } = \mathbf{32\cdot \sqrt{3} }[/tex]
The perimeter of the hexagon, P = 32·√3 cm
The apothem of a regular hexagon bisects the angle formed by the two
adjacent radii of the hexagon at the center of the hexagon.
The angle formed by the hexagon at the center = 60°
Therefore;
- The measure of the angle formed by the radius and the apothem is 30°.
The side length of the hexagon, s ≈ 9.24 cm
The interior angles of an hexagon are;
[tex]\dfrac{(6 - 2) \times 180^{\circ}}{6} = 120^{\circ}[/tex]
Each radius bisects an interior angle, therefore;
Angle formed by a radius and an adjacent side = 60°
Which gives;
The triangle formed by two radii and a side = An equilateral triangle
Side lengths of an equilateral triangle are equal, therefore;
- The radius and side length of a regular hexagon (sides of an equilateral triangle) are equal.
The area, A, of the hexagon is given as follows;
[tex]A = \mathbf{ \dfrac{1}{2} \times P \times a}[/tex]
Which gives;
[tex]A = \dfrac{1}{2} \times 32 \cdot \sqrt{3} \times 8 \approx \mathbf{ 221.07}[/tex]
Therefore;
- The area of the hexagon is about 221.7 square cm
Learn more about a regular hexagon here:
https://brainly.com/question/21502832
