Answer:
17.661cm
Step-by-step explanation:
First, we need to find the length of arc AB here. Then the perimeter of the sector will be given by:
Perimeter = Length of (Arc AB + OA + OB)
For finding the length of AB arc, we may use the formula:
[tex]\theta=\dfrac{l}{r}[/tex]
where,
[tex]\theta[/tex] = angle subtended at the center,
[tex]l[/tex] = length of arc,
and [tex]r[/tex] = radius of circle/sector.
In this case,
[tex]\theta=80^\circ[/tex], [tex]l=[/tex] ?, [tex]r=[/tex] 5.2cm
Before anything else, let's convert [tex]\bold{\theta}[/tex] to radians. It can be done using the following identity:
[tex]\pi^c=180^\circ\\\\\text{or, }1^\circ=\bigg(\dfrac{\pi}{180}\bigg)^c\\\\\text{or, }80^\circ=\bigg(\dfrac{\pi}{180}\times80\bigg)^c\\\\\text{or, }80^\circ=\bigg(\dfrac{4\pi}{9}\bigg)^c\\\\\boxed{\therefore\ \theta=\bigg(\dfrac{4\pi}{9}\bigg)^c}[/tex]
Now we will finally use [tex]\theta=\dfrac{l}{r}[/tex] formula to find the length of arc[tex](l).[/tex]
[tex]\dfrac{4\pi}{9}=\dfrac{l}{5.2}\\\\\text{or, }l=\dfrac{20.8\pi}{9}\\\\\text{or, }l=7.261\text{cm}\ \ \ (\text{to three significant figures.})[/tex]
So, the length of arc AB is approximately 7.261cm.
Finally,
perimeter of the sector = Length of (arc AB + OA +OB) = 7.261 + 5.2 + 5.2 = 17.661cm
