Question 1067518
<pre><b>


{{{drawing(400,4800/19,-1,8.5,-1,5,
line(0,0,3.247642976,2.692803301),
locate(3.5,0,7.5), locate(5.3,1.7,5),
locate(2,3.8,3),
locate(.7,.5,alpha),locate(.35,1.1,alpha),
locate(2.6,2.8,beta),locate(2.8,2.4,pi-beta),
locate(.24,2.5,x),

red(arc(0,0,1.5,-1.5,0,40),arc(0,0,1.8,-1.8,40,80)

),
triangle(0,0,7.5,0,5/6,4.22166387))}}}

By the law of sines in the upper triangle,

       {{{3/sin(alpha)}}}{{{""=""}}}{{{x/sin(beta)}}}

eq. 1  {{{3*sin(beta)}}}{{{""=""}}}{{{x*sin(alpha)}}}

Also by the law of sines in the lower triangle,

       {{{5/sin(alpha)}}}{{{""=""}}}{{{7.5/sin(pi-beta)}}}

       {{{5*sin(pi-beta)}}}{{{""=""}}}{{{7.5*sin(alpha)}}}

And since {{{beta}}} and {{{pi-beta}}} are supplementary,
they have the same sine:

eq. 2  {{{5*sin(beta)}}}{{{""=""}}}{{{7.5*sin(alpha)}}}

Dividing equals by equals using eq. 1 and eq. 2:

       {{{3/5}}}{{{""=""}}}{{{x/7.5}}}

       {{{5x}}}{{{""=""}}}{{{3*7.5}}}
      
       {{{5x}}}{{{""=""}}}{{{22.5}}}
 
        {{{x}}}{{{""=""}}}{{{4.5}}}

That's only one solution.  There is another solution.
Here is the drawing:

{{{drawing(400,320,-1,11,-1,8.6, locate(1.5,.6,alpha),
locate(1.45,1.15,alpha), locate(3.5,0,7.5),
locate(8.05,1.52,3),locate(9.2,5,5), locate(8.1,3.5,beta),
locate(7,2.4,pi-beta),locate(4.5,4.2,x),
triangle(0,0,7.5,0,9.9,7.631513611), line(0,0,8.4,2.861817604),
red(arc(0,0,3,-3,0,19),arc(0,0,3.3,-3.3,19,38))




  )}}}

You find the other solution.  It's done the same way as the 
first solution.  

Edwin</pre></b>