What is the length of the shorter diagonal of the parallelogram if the lengths of the two adjacent sides of parallelogram ABCD are 8 and 14 units respectively and if the degree measure of the included angle is 60º?

1 Answer
Dec 12, 2014

Its typically helpful to start with a drawing. Here is #ABCD# as given by the problem.

Geogebra

We are looking for the length of the shorter diagonal, which is segment #BD#. This segment forms a triangle with the two known sides. Since we know two sides and the angel connecting them, we can use the law of cosines to solve for the unknown segment.

http://mathworld.wolfram.com/LawofCosines.html

The law of cosines tells us that #c^2 = a^2 + b^2 - 2ab cos(C)# for the triangle labeled above. If we choose our two known sides for #a# and #b# and our known angle for #C#, we can solve for the length of the diagonal, #c#.

#c^2 = 14^2 + 8^2 - 2 ( 14 )(8)cos(60^o) ~~ 12.17#