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?

Logic Apr 19, 2019

#1**+1 **

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

CPhill Apr 20, 2019