answer: this problem has only trivial solutions, with all points on the same line. x=5 or x=7/5.
step-by-step explanation:
there are three triangles, and three unknowns: x, y=length of middle segment, and z=cos(angles marked with red arc).
we need
![\cos(2\gamma)=(2\cos(\gamma)^2-1)](/tex.php?f=\cos(2\gamma)=(2\cos(\gamma)^2-1))
and law of cosines,
![c^2=a^2+b^2-2ab\cos(\gamma)](/tex.php?f=c^2=a^2+b^2-2ab\cos(\gamma))
the three equations are
![(2x-1)^2=9^2+y^2-2(9yz)](/tex.php?f=(2x-1)^2=9^2+y^2-2(9yz))
![(3x)^2=15^2+y^2-2(15yz)](/tex.php?f=(3x)^2=15^2+y^2-2(15yz))
![(3x+2x-1)^2=9^2+15^2-2(9\times15(2z^2-1))](/tex.php?f=(3x+2x-1)^2=9^2+15^2-2(9\times15(2z^2-1)))
see attached image for solution.
![What is the value of x? enter your answer in the box. x = ​​](/tpl/images/02/06/fUtMCs18IkARoQl5.jpg)