A triangle has sides A, B, and C. Sides A and B are of lengths #4# and #2#, respectively, and the angle between A and B is #(3pi)/4 #. What is the length of side C?

1 Answer
Aug 1, 2016

Answer:

#abs(C)=color(green)(2sqrt(5+2sqrt(2)))#

Explanation:

If #c# is the angle opposite side #C#
then #c=(3pi)/4# (given)

and by the Law of Cosines
#color(white)("XXX")C^2=A^2+B^2-2ABcos(c)#

In this case
#color(white)("XXX")C^2=4^2+2^2-2 * 4 * 2 * (-sqrt(2)/2)#

#color(white)("XXXX")=20+8sqrt(2)#

#color(white)("XXXX")=2^2(5+2sqrt(2))#

#rArr#
#color(white)("XXX")abs(C)=2sqrt(5+2sqrt(2))#

(or using a calculator to evaluate an approximation)
#color(white)("XXX")abs(C)~~5.595865#