Two sides of an acute triangle are 8 and 15. How many possible lengths are there for the third side, if it is a positive integer?
We have the following inequalities
8 + 15 > x where x is the unknown side
23 > x
x < 23 and
8 + x > 15
x > 7
However....since the triangle is acute....we know that an 8 - 15 - 17 triangle is a right triangle
So....the remaining side, x, must be < 17 [ any integer > 17 but < 23 will produce an obtuse triangle ]
So....the possible side lengths are integers between 7 and 17
However....we must even restrict this interval to integers greater than 12 and < 17
The reason for this is that integer side lengths for x from 8 to 12 inclusive also produce obtuse triangles
So....4 possible acute triangles