Given the following triangle side lengths, identify the triangle as acute, right or obtuse. show your work.
a. 3in, 4in, 5 in

b. 5in, 6in, 7in

c. 8in, 9in, 12in

Let "a", "b" and "с"  be sides of the triangle ("с" is the longest side).
The triangle will be:

right if       a² + b² = c²
аcute if     a² + b² > c²    
obtuse if   a² + b² < c²    

a.
a=3, b=4 and c=5

a² + b² = 3² + 4² = 9 + 16 = 25   and   c² = 5² = 25

25 = 25   ⇒  right triangle.

b.
a=5, b=6 and c=7

a² + b² = 5² + 6² = 25 + 36 = 61   and   c² = 7² = 49

61 > 49   ⇒  аcute triangle.

c.
a=8, b=9 and c=12

a² + b² = 8² + 9² = 64 + 81 = 145   and   c² = 12² = 144

145 > 144   ⇒  аcute triangle.