A triangle has two sides of length 19 and 3. What is the largest possible whole number length for the third side?

Explanation:

Consider a triangle with side lengths a,b,c such that a < b < c

a = 3 and b = 19 are the two known sides

Let c be the missing side length

The missing side has the property that...

b-a < c < b+a

19-3 < c < 19+3

16 < c < 22

Meaning that if c is a whole number then it can take on any of these values {17, 18, 19, 20, 21}

So c = 21 is the largest possible third side, where only whole number sides are considered.

We can't have a third side of 22 because the two other sides add to a+b = 3+19 = 22 and a triangle wouldn't form (instead only a straight line would). I recommend cutting out strings of paper trying this out yourself.

Side note: The inequality b-a < c < b+a is a variation of the triangle inequality theorem.

c

step-by-step explanation: