Question

The length of two sides of triangle are 4cm and 5cm. Between what two

measures should the length of the third side fall?

Answer 1

1 and 9

Explanation:

By the Triangle Inequality Theorem, the sum of two side lengths of a triangle is always greater than the length of the third side.

In other words, in a triangle with side lengths
`a, b,` and
`c,` we always have
`a+b > c.`

Applying this to this question, the other side length,
`x,` must satisfy the following inequalities:

`4+5 > x,\\x+5 > 4,\\x+4 > 5.`

Solving these inequalities gives

`x < 9,\\x > -1,\\x > 1.`

Combining these solutions, we have
`1 < x < 9.` Therefore, the length of the third side falls between 1 and 9.