Question 763795
<pre>
We want the shorter diagonal BD, 
but first we must calculate the longer diagonal, 
the green line AC below:

{{{drawing(200,1750/3, -.6,.6,-2,1.5,
locate(-.55,.05,B),locate(.5,.05,D),
green(line(0,-1.5299,0,1.1048)),
locate(-.02,1.2,A),locate(-.02,-1.53,C),
line(-.4684,0,0,1.1048),locate(-.43,.05,"140°"),
line(.4684,0,0,1.1048), locate(-.42,.6,1.2),locate(-.4,-.75,1.6),
line(-.4684,0,0,-1.5299),arc(-.4684,0,.5,-.5,287,427),
line(.4684,0,0,-1.5299) )}}}

We use the law of cosines on &#916;ABC since this is the SAS case:

AC² = AB² + BC² - 2·AC·BC·cos(&#8736;ABC)

AC² = 1.2² + 1.6² - 2(1.2)(1.6)cos(140°)

 AC = 2.634693656 m

We use the law of sines to find &#8736;BAC

{{{BC/sin(BAC)}}}{{{""=""}}}{{{AC/sin(B)}}}

{{{1.6/sin(BAC)}}}{{{""=""}}}{{{2.634693656/sin("140°")}}}

Cross multiply:

(2.634693656)sin(&#8736;BAC) = (1.6)sin(140°)

             sin(&#8736;BAC) = {{{(1.6sin("140°"))/2.634693656}}}

                  &#8736;BAC = 22.97645649°

Now that we have &#8736;BAC we draw half the shorter diagonal BE, 
which is perpendicular to the longer diagonal AC, in red:

{{{drawing(200,1750/3, -.6,.6,-2,1.5,
locate(-.55,.05,B),locate(.5,.05,D),locate(0.02,.05,E),
green(line(0,-1.5299,0,1.1048)), red(line(-.4684,0,0,0)),
locate(-.02,1.2,A),locate(-.02,-1.53,C),
line(-.4684,0,0,1.1048),rectangle(-.05,0,0,.05),
line(.4684,0,0,1.1048), locate(-.42,.6,1.2),locate(-.4,-.75,1.6),
line(-.4684,0,0,-1.5299),arc(0,1.1048,.5,-.5,247,270),
line(.4684,0,0,-1.5299) )}}}

Since &#916;ABE is a right triangle:

{{{(BE)/AB}}} = {{{(opposite)/(hypotenuse)}}} = sin(BAC)

BE = AB·sin(BAC)

BE = 1.2·sin(22.97645649°)

BE = 0.4684234191 m

So the entire shorter diagonal, BD, 

{{{drawing(200,1750/3, -.6,.6,-2,1.5,
locate(-.55,.05,B),locate(.5,.05,D),locate(0.02,.1,E),
green(line(0,-1.5299,0,1.1048)), red(line(-.4684,0,.4684,0)),
locate(-.02,1.2,A),locate(-.02,-1.53,C),
line(-.4684,0,0,1.1048),rectangle(-.05,0,0,.05),
line(.4684,0,0,1.1048), locate(-.42,.6,1.2),locate(-.4,-.75,1.6),
line(-.4684,0,0,-1.5299),arc(0,1.1048,.5,-.5,247,270),
line(.4684,0,0,-1.5299) )}}}

is twice the length of BE, so

BD = 2·BE = 2(0.4684234191) = 0.9368468381 m 

Rounded to tenths:  0.9 m

Edwin</pre>