Question 701045
If the length of the rectangle is two times the width of the rectangle, the diagonal does not split the rectangle into 30-60-90 triangles.
The diagonal forms two congruent right triangles, but the smaller angle measures less than {{{30^o}}}.
{{{drawing(300,150,-1,15,-1,7,
rectangle(0,0,13.416,6.708),
green(line(0,0,13.416,6.708)),
locate(2,1,alpha),locate(6,4,15),
locate(6,0,2x),locate(13.5,4,x)
)}}}
{{{tan(alpha)=x/(2x)=1/2}}} --> {{{alpha=26.565^o}}} (rounded)
{{{x=15*sin(alpha)=15*0.447214=6.7}}} in (rounded)
 
A diagonal splits a rectangle into two congruent 30-60-90 triangles, if, and only if, the diagonal is twice as long as the width of the rectangle.
In such a rectangle,
{{{drawing(275,210,-0.5,9,-1,6,
rectangle(0,0,8.66,5),
green(line(0,0,8.66,5)),
locate(1.5,1,30^o),locate(4,3,15),
locate(4.5,0,L),locate(8.75,3,W)
)}}}
{{{sin(30^o)=w/15=1/2}}} --> {{{W=(1/2)*15=7.5}}} in
{{{L=15*cos(30^o)=15*0.866=12.99}}} in (rounded)