The sum of the lengths of any two sides of a triangle

Question

The sum of the lengths of any two sides of a triangle must be greater than the third side. If a triangle has one side that is 8 inches and a second side that is 4 inches less than twice the third side, what are the possible lengths for the second and third sides?

Answer 1

 Let's call the three sides of the triangle a, b, and c.

Then, we have:

 a = 8

 b = 2c - 4

 

From the triangle inequality, there are two possibilities:

Either

a + c > b

or

b + c > a

Case (1):

 a + c > b

 8 + c > 2c - 4

 12 > c

 

Case (2):

 b + c > a

 3c - 4 > 8

 3c > 12

 c > 4

 

Therefore:

 4 < c < 12

4 < b < 20