Hi Jennifer O
First did you draw your figure? I would recommend doing so.
Secondly, I have a question, are you referring to a Regular Nonagon, a nine sided figure with nine interior angles of 140 degrees each and nine central angles of 40 degrees each, that form 9 Isosceles Triangles?
If so, then the leg of each isosceles triangle splits the interior angles into base angles of 70 degrees each. You are given the length from a vertex to the center of the Nonagon as 8cm this length represents the size of the legs of the Isosceles Triangles. You can use the Trigonometric Ratios, or if you are familiar with the law of Sines you can look that up use it, finally you can also look up the Perimeter of a Regular Nonagon.
You should draw one of the Isosceles Triangles with base angles each of 70 degrees and legs each of 8 cm, and central angle of 40 degrees to see what you are dealing with. If you draw a height through the center of your central angle (splitting your central angle into two 20 degree angles) to the base of the unknown side you get two Right Triangles each with angles of 20, 70, and 90 degrees and 8 cm for each hypotenuse. You can calculate half of the base(half of your missing side) directly using Trigonometric Ratios
Let Half of the missing side = x
sin(20) = x/8
8 sin(20) =x
But you really need 2x to get the entire missing side so substituting
2(8sin(20)) = 2x
2(8sin(20)) = 5.472
5.472 is one of the sides of the Regular Nonagon but you need the Perimeter so that means all 9 sides or
9(5.472) = 49.251 cm
Of course there are other ways to approach the problem with known formulas, Trigonometry, even area but
I hope this helps.
