Triangle abd is a right-angled triangle, with ∠abd = 90◦ . the point c lies on side bd. cd has length 20 cm. ∠cda = 35.8 ◦ and ∠bca = 50.5 ◦ . find the length of ca, giving your answer to two significant figures

The length is 46.10 cm.

Since ∠BCA = 50.5°, ∠ACD = 180-50.5 =129.5°.  This means that ∠CAD = 180-(129.5+35.8) = 14.7°.  We can now use the law of sines;

sin 14.7/20 = sin 35.8/x

Cross multiply:
x*sin 14.7 = 20*sin 35.8

Divide both sides by sin 14.7:
x = (20*sin 35.8)/(sin 14.7) = 46.10