Question

A triangle has two sides of lengths 4 and 5. What value could the length of the third side be?

Answer 1

The length of the third side could be all real numbers greater than 1 unit and less than 9 units

Explanation:

we know that

The Triangle Inequality Theorem, states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side

so

Applying the triangle inequality theorem

Let

x ----> the length of the third side

1) 4+5 > x

9 > x

Rewrite

x < 9 units

2) 4+x > 5

x > 5-4

x > 1 units

therefore

The solution for the third side is the interval -----> (1,9)

All real numbers greater than 1 unit and less than 9 units

therefore

The length of the third side could be all real numbers greater than 1 unit and less than 9 units

Answer 2

2, 3, 4, 5, 6, 7 or 8.

Explanation:

We know that the sum of two sides on a triangle should ALWAYS be greater than the third side. Then we have:

5-4 < x < 5 + 4

1 < x < 9

Therefore, the lenght of the third side could be any number between 1 and 9. If the lenght of the third side is an integrer, then the lenght could be:

2, 3, 4, 5, 6, 7 or 8.