Interesting Trig Question

[Pearl07]

In triangle ABC, side BC is the average of the other two sides. If cos C = AB/AC , find the numerical value of cos C.

There are no numerical values given, so I'm stuck. How would I approach this? Would I use the law of cosines?

 

[Dr.Peterson]

With no numerical values, I would just call the sides a, b, c, and write some equations. One will be a = (b+c)/2, the average of the other two sides. Another will express cos C = AB/AC, and a third will be the Law of Cosines. (You can quickly replace cos(C) with c/b so you're down to three variables.)

Then solve, keeping in mind that you don't need a value for any of the variables, just for the ratio c/b.