Rhombus problem?

[stateofpeace]

It's the first problem on this sheet:

http://www.jmap.org/StaticFiles/PDF...rmal_and_Formal_Proofs/Drills/PR_G.G.39_7.pdf

[Well technically it's the first three, but once I understand the first one I can do the other three...]

I know the properties of a rhombus, so obviously all of the sides are 24, the angles created by the diagonals are 90 degrees, angles A and C are 60 degrees, angles B and D are 120. What I'm not sure is how to apply the properties in order to solve this problem. Just need a hint as to what to do...

 

[stapel]

Have you done trig at all, so you could the sine ratio to the right triangle AED?

 

[lPing7]

AED is a 90, 60, 30 triangle
Therefore ratio of sides to side to hypotenuse is
1 is to sqrt3 is to 2
This can be proved using trig

 

[stateofpeace]

Yeah, trig ratios were the only way I could think to solve it, but this is geometry and our teacher never taught us that, so I wasn't sure if I should do that. Is there a 'geometric' way to do it? Otherwise I'll go with the trig. Thanks for the direction so far!

 

[stateofpeace]

WAIT I found a 'geometric' way to solve it:

If the measure of angle BAC is 30, then angles A and C are 60 because diagonals bisect angles. Then B and D are 120. Using this knowledge, triangle DCB is an equilateral triangle, so all the angles are 60 degrees, and all of the sides are 24. They want DE. Since the side is 24 and diagonals bisect one another, DE would therefore be 12.

Is there a cleaner way of doing this, or is the cleanest way only through trig? ;__;