SOLUTION: A triangle has one side that is 8" and another side that is 13". What I'd the minimum length of the third side? What is the maximum length of the third side?

Question 1205187: A triangle has one side that is 8" and another side that is 13". What I'd the minimum length of the third side? What is the maximum length of the third side?
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

I'll be using the Triangle Inequality Theorem.
More info is found here
https://www.algebra.com/algebra/homework/Triangles/triangle-inequality-theorem.lesson
Specifically I'll use the extension of that theorem mentioned after example 2.

If we know two sides of a triangle a and b, then possible values for the third side c is found through the inequality:
b-a < c < b+a
where b > a

In this case
a = 8
b = 13
So,
b-a < c < b+a
13-8 < c < 13+8
5 < c < 21

Side c is between 5 inches and 21 inches, excluding both endpoints.
Something like c = 5.1 is valid, but c = 5 is not.
Technically there is no smallest value for c because we could have the following
c = 5.1
c = 5.01
c = 5.001
c = 5.0001
and so on
We can approach 5 but not actually arrive there.

If we limit c to just the integers, then c = 6 would be the smallest side possible.
Otherwise, there is no smallest value.

The same can be said about the other end of the spectrum as well.
There is no largest value if c is a real number (since we could have c = 20.9 or c = 20.99 or c = 20.999 and so on).
But if c was an integer, then c = 20 is the largest possible.

Answer by ikleyn(52792)   (Show Source): You can put this solution on YOUR website!
.

In this problem, both questions are posed INCORRECTLY.

As other tutor just explained you, there is NO maximum length for the third side in this case.

Similarly, there is NO minimum length for the third side in this problem.

A correct mathematical wording is "what is the upper bound for the length of the third side?", and

"what is the lower bound for the length of the third side?".

                        Do you create Math problems on your own ?
        From reading your post, it is the only conclusion which I can make.

To create Math problems properly, a certain mathematical training is needed . . .

12 years of diligent study of mathematics at school, then 4-5 years at university, plus 5 years of teaching practice.

It can be shorter - but worse . . .

It can be much shorter - but much worse . . .

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