Question 1172080
<pre>

{{{drawing(400,4400/19,-6,13,-2,9,

line(0,0,12,0), line(12,0,6.700543302,7.856767654),

line(6.700543302,7.856767654,-5.299456698,7.856767654),
line(-5.299456698,7.856767654,0,0), locate(0,1.4,124^o),
locate(3,5,19),locate(-2.8,3.6,x),
line(-5.299456698,7.856767654,12,0),locate(4.7,0,12) )}}}

Use the law of cosines:

{{{19^2=x^2+12^2-2(12)(x)cos(124^o)}}}

{{{361=x^2+144-(24x)(-0.5591929035)}}}

{{{361=x^2+144+13.42062968x}}}

{{{x^2+144+13.42062968x-347.5793703=0}}}

Solve by the quadratic formula:

{{{x=9.476974162}}}

We draw it with the other diagonal:

{{{drawing(400,4400/19,-6,13,-2,9,

line(0,0,12,0), line(12,0,6.700543302,7.856767654),
locate(-4.6,8,56^o),
line(6.700543302,7.856767654,-5.299456698,7.856767654),
line(-5.299456698,7.856767654,0,0), locate(0,1.4,124^o),
locate(-5,3.6,9.476974162),locate(3,5,y), locate(0,8.7,12),
line(6.700543302,7.856767654,0,0),locate(4.7,0,12) )}}}

{{{y^2=9.476974162^2+12^2-2(9.476974162)(12)cos(56^o)}}}

{{{y^2=106.626}}}

{{{y = 10.326}}}  <--answer (length of other diagonal.)

Edwin</pre>