Answer:
(A). The perimeter of the octagon is greater than that of the hexagon.
Step-by-step explanation:
Since, hexagon consists of 6 sides and 6 angles, thus the measure of one angle of the hexagon will be=[tex]\frac{(n-2){\timeS}180^{\circ}}{6}[/tex]
=[tex]\frac{(6-2){\timeS}180^{\circ}}{6}[/tex]
=[tex]\frac{(4){\timeS}180^{\circ}}{6}[/tex]
=[tex]120^{\circ}[/tex]
Now, since MQ is the angle bisector of the one of the angle of the hexagon, therefore ∠QMP=60°.
Now, from ΔQMP. we have
[tex]\frac{MP}{MQ}=cos60^{\circ}[/tex]
[tex]MP=\frac{1}{2}[/tex]
Thus, the perimeter of the hexagon is:
[tex]P=12{\times}MP[/tex]
[tex]P=12{\times}\frac{1}{2}[/tex]
[tex]P=6 units[/tex]
Thus, the perimeter of hexagon is 6 units.
Also, Since, octagon consists of 8 sides and 8 angles, thus the measure of one angle of the octagon will be=[tex]\frac{(n-2){\timeS}180^{\circ}}{8}[/tex]
=[tex]\frac{(8-2){\timeS}180^{\circ}}{8}[/tex]
=[tex]\frac{(6){\timeS}180^{\circ}}{8}[/tex]
=[tex]135^{\circ}[/tex]
Now, since AP is the angle bisector of the one of the angle of the octagon, therefore [tex]{\angle}PAC=cos\frac{135}{2}[/tex].
From ΔAPC, we have
[tex]AC=cos\frac{135}{2}[/tex]
Now, Perimeter of octagon is:
[tex]P=16{\times}cos\frac{135}{2}[/tex]
[tex]P=16{\times}0.382[/tex]
[tex]P=6.122 units[/tex]
Thus, the perimeter of octagon is 6.122 units.
Now, the perimeter of octagon is greater than perimeter of the hexagon, thus option A is correct that is The perimeter of the octagon is greater than that of the hexagon.
