I need help answering how long the side is. Please explain clearly if you can. :)

9514 1404 393

  12.44 cm

Step-by-step explanation:

Since there are so many layers to this problem, we must keep full calculator precision in all calculations. The numbers will be reported here to 5 significant figures, but 12 or 16 significant figures are appropriate for intermediate calculations. A spreadsheet may be useful for computing and documenting results.

As noted in your second attachment, SOH CAH TOA and the Pythagorean theorem will be used. Variations on these will give unknown sides in terms of known sides and angles. Generally, we will be working from the bottom up.

Angles will be signified by degrees. Lengths are in cm, and not marked with a unit.

__

Green hypotenuse = (8)/cos(35°) = 9.7662

Yellow hypotenuse = Green hypotenuse -2.5 +3.6 = 10.866

Yellow long leg = Yellow hypotenuse × cos(14°) = 10.543

Orange hypotenuse = Yellow long leg -2.1 +2.9 = 11.343

Orange long leg = √(11.343² -3.8²) = 10.688

Blue long leg = Orange long leg -3.2 = 7.4880

Blue short leg = Blue long leg × tan(42°) = 6.7422

adjacent clear hypotenuse = Blue short leg/sin(37°) = 11.203

Dark yellow hypotenuse = adjacent clear hypotenuse +4.08 = 15.283

Dark yellow long leg = Dark yellow hypotenuse × sin(57°) = 12.818

adjacent clear long leg = Dark yellow long leg - 2.2 = 10.618

adjacent clear hypotenuse = √(10.618² +1.7²) = 10.753 = Red long leg

Red hypotenuse = Red long leg/cos(20°) = 11.443

Pink hypotenuse = Red hypotenuse +1.5 = 12.943

how long is this side = Pink hypotenuse × sin(74°) = 12.441

how long is this side = 12.44 cm

Additional comment

It looks like there is not quite enough information to solve the purple and lime triangles at the left side of the figure. Those are irrelevant to the question asked here, so that is of no consequence.