anglegeometrypolygons
I need to find the vertex coordinates of a pentagon given that one of the coordinates is $(1,0)$ and that the centre of the pentagon is located at $(0,0)$?
This is to be done only by using geometry (and trigonometry), without using complex numbers. This can possibly be done by splitting sectors into triangles and using trigonometry with the $72°$ and $108°$ angles, however I am unsure how to proceed.
Let $B = (1, 0)$ and center $(0,0)$:
Let $C(c_x,c_y)$. Then, we have two equations about $c_x,c_y$ :
$$\vec{BA}\cdot\vec{BC}=0\iff (a_x-b_x)(c_x-b_x)+(a_y-b_y)(c_y-b_y)=0$$$$BC=L\iff (c_x-b_x)^2+(c_y-b_y)^2=L^2$$ If you have $c_x\lt b_x$, then you can get $c_x,c_y$ by solving these.
Each pentagon consists of an equilateral triangle and a rhombus, see figure (and the Note at the end) 
To obtain a desired tiling, you can consider the length of each side equal $1.$
The center of the configuration is in $0.$ Starting with two points (here $0$ and $1$), any further point is obtained by a rotation of a point we already have. With the use of complex numbers has a rotation a simple formula $$z'-z_0=e^{i\varphi}(z-z_0),$$ where $z_0$ is ordinate of the center of the rotation, $z$ is that of the point you want to rotate, and $z'$ is the obtained image of $z.$ Complex ordinates of some points are enclosed.
Note added: there is a typo in the picture, the most right point is obtained by a clockwise rotation (negative angle), the right term is $e^{-i10\pi/18}.$
Best Answer
Draw the axes and center $O=(0,0)$
From $B$, drop perpendicular $BP$ onto x-axis and look at the right triangle formed. Point $B$ will have abscissa of length $OP = OB\cos \theta_B$ and ordinate equal to length $BP = OB\sin \theta_B$. In addition we need to be careful with signs. Since $B$ is in first quadrant, its coordinates will have signs $(+,+)$. Conclude $$B=(+OB\cos \theta_B, +OB\sin \theta_B)$$
Now resorting to geometry, $\theta_B = \angle AOX - \angle AOB=90^\circ - 72^\circ = 18^\circ$. As $OB=1$, $$B=(+\cos 18^\circ, +\sin 18^\circ)$$
Similarly you can find coordinates of $C,D,E$.
Remark :
I found coordinates of $B$ for the condition $A=(0,1)$, which was mentioned in an earlier version of this post. However if it is given $B=(1,0)$, then question becomes even easier; same concept applies. Find the angle corresponding to each vertex and then append proper signs to abscissa and ordinates.