Length of a line in a regular hexagon

Question

The picture belongs to a regular hexagon. I don't understand that if a line is drawn from $E$ to $C$, how its length equals to $|KL|$ and how it is parallel to $|KL|$?

Would you mind drawing its explanation?

Answer 1

The two lines are not parallel or have the same length. We know the interior angle of a regular hexagon is $120^\circ$ and that the angles ECB and CEF are right ones.

The line EC has a length of $6\sqrt3$. This can be calculated by the cosine rule: $a^2=c^2+b^2 -2bc\cos(a)$ substituting: $a^2 = 72+72\cos(120)$ so $a = 6\sqrt3$.

We can find the length of KL by finding the hypotenuse of a right angled triangle with base $6\sqrt3$ nd side length 3. By Pythagoras' theorem this gives us $a^2=3^2+(6\sqrt3)^2$ -o $a^2 = 9+108 = 117$ so $a = 3\sqrt13$

We know the base of the triangle is $6\sqrt3$ because the lines CB and EF are parallel. We can find the height of this triangle by taking the difference of 4(the length LC) and 1(the length KE) So in conclusion, the lines are not parallel of of the same length. Line KL is $3\sqrt13$ units long while EC is $6\sqrt3$ units long. I hope this was of some assistance.