find the length of the side not given to a triangle. l=39, l=31

Question

what is the length not given

Answer 1

If the side lengths are a=39, b=31 and c (unknown), then the Triangle Rule is:

a+b>c; a+c>b; b+c>a, so:

39+31>c, c<70; 39+c>31; 31+c>39, c>8.

Therefore 8<c<70, meaning c is between 8 and 70 in length.

To find the length of c exactly, we need the angle C between a and b, then the Cosine Rule applies:

c=√(a²+b²-2abcosC)=√(1521+961-2418cosC)=√(2482-2418cosC)