Complicated trigonometry problem

[Natasha1]

Could someone check this for me and help answering the questions at the end

The 3 angles around point 6 are 120 degrees each. This is also the case for point 8 which is linked to point 2, 3 and 7.

Now the 3 angles formed by the lines leaving point 7 are also 120 degrees.

If the distance between point 1 and 2, 2 and 3, 3 and 4, 4 and 5 is 1 cm.

5,6,4 forms an isosceles triangle and so does 2,8,3 triangle.

What is the total distance of the lines?

My answer:

Now, let's start by dropping a vertical line segment 1-7 or a. At point 7, we branch into two lines, one of whom is 7-6 or b. From point 6, we branch into 6-5 or c, and 6-4 or d.

Now, segment a bisects the 108o angle at point 1.
Lets call the side 5-1, e.
In quadrilateral abce, angle 6-5-1 = 360 - 54 - 120 - 120 = 66o.

Lets call side 4-5, f.
In triangle cdf,
Angle 6-5-4 = 108 - 66 = 42o.
So, angle 6-4-5 = 180 - 120 - 42 = 18o.
(This triangle therefore is not isosceles.)

Using the sine rule,
f/sin(120o) = c/sin(18o) = d/sin(42o)
Since f = 1 cm,
c = 1*sin(18o)/sin(120o) = 0.356822 cm.
d = 1*sin(42o)/sin(120o) = 0.772645 cm.

Lets call line segment 1-6, g.
In triangle gce,
Using the cosine rule,
g2 = (c)2 + (e)2 - 2(c)(e)cos(66o)
g2 = (0.356822)2 + (1)2 - 2(0.356822)(1)cos(66o)
g = 0.914908 cm

Using the sine rule,
c/sin (angle 5-1-6) = g/sin(66o)
(0.356822)/sin(angle 5-1-6) = (0.914908)/sin(66o)
So, angle 5-1-6 = arcsin[(0.356822)*sin(66o)/(0.914908)] = 20.872561o

In triangle abg,
g = 0.914908 cm.
Angle 6-1-7 = 54 - 20.872561 = 33.127439o
Angle 7-6-1 = 180 - 33.127439 -120 = 26.872561o

Using the sine rule,
(0.914908)/sin(120o) = a/sin(26.872561o) = b/sin(33.127439o)
So,
a = (0.914908)*sin(26.872561deg)/sin(120o) = 0.477521 cm
b = (0.914908)*sin(33.127439deg)/sin(120o) = 0.577350 cm

Therefore, the total length of the shortest way is:
= a + 2 (b + c + d)
= 0.477521 + 2 (0.577350 +0.356822 +0.772645) = 3.891155 cm.

How can I find the total length in the hexagon (see attached picture)

 

[StatusX]

Your description of the problem doesn't seem to match the pictures. For example, I only see one line connected to point 6, not three. And what do you mean by finding the total length of the lines? Are supposed to find the length of every line in the diagram and then add these up? Please try to describe the problem more clearly, and maybe focus only on the parts that are giving you the most trouble.