Aparallelogram has adjacent sides that are 11.0 cm and 15.0 cm long. the angle between the sides is 50°. determine the length of the shorter diagonal to the nearest 10th of a centimetre. and explain your answer thoroughly,,

Sine (50) = height / 11
height = sine (50) * 11
height = .76604 * 11
height = 8.42644
Line AE^2 = 11^2 - 8.42644^2
Line AE^2 = 121 - 71.0048910736
Line AE^2 = 49.9951089264
Line AE = 7.0707219523
Line EB = 15 -7.0707219523
Line EB = 7.9292780477
(Diagonal "p")^2 = 7.93^2 + 8.43^2
(Diagonal "p")^2 = 62.8849 +  71.0649
(Diagonal "p")^2 = 133.9498
Diagonal "p" = 11.574

Here's a parallelogram calculator to check the result:
http://www.1728.org/quadpar.htm