EXPLANATION
With the length of the apothem, we can get the length of the side by applying the following relationship:
[tex]a=x\sqrt{3}[/tex]
Where a represents the apothem and x represents a half of the length of the side.
Plugging in the terms:
[tex]8=x\sqrt{3}[/tex]
Isolating the value of x:
[tex]x=\frac{8}{\sqrt{3}}[/tex]
Simplifying:
[tex]x=\frac{8\sqrt{3}}{3}\approx4.62[/tex]
Now, the side length is equivalent to two times the value of x:
[tex]side\text{ length=s=2x=2*4.6=9.2cm}[/tex]
The side length is 9.2cm
c) Therefore, the third statement is not true.
a) Multiplying the side length by 6 give us the perimeter:
[tex]9.2*6=55.2cm[/tex]
Therefore, the perimeter is 55.2, not 48cm. The first statement is not true.
d) The fourth statement is true, in a regular hexagon, the radius equals the side length.
e) Area of the hexagon:
The area of the hexagon can be obtained by applying the following equation:
[tex]Area_{regular\text{ hexagon}}=\frac{1}{2}*Perimeter*apothem[/tex]
Plugging in the known terms into the expression:
[tex]Area_{regular\text{ hexagon}}=\frac{1}{2}*55.2*9.2=253.92[/tex]
Since the area is approximately equal to 253.92, the last statement is not true.
In conclusion, the solutions are:
a) NOT TRUE
b) TRUE
c) NOT TRUE
d) TRUE
e) NOT TRUE
