Question

A triangle has side lengths of 24 centimeters and 35 centimeters. The third side has a length of x centimeters.
Complete the inequality to show all the possible values of x. Enter the answer in each box.

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Answer 1

To determine the possible values of the third side \( x \) of a triangle with side lengths \( 24 \) cm and \( 35 \) cm, we can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.

We need to apply the triangle inequalities:
1. \( 24 + 35 > x \)
2. \( 24 + x > 35 \)
3. \( 35 + x > 24 \)

Let's solve these inequalities one by one:

1. From \( 24 + 35 > x \):
\[
59 > x \quad \text{or} \quad x < 59
\]

2. From \( 24 + x > 35 \):
\[
x > 35 - 24 \quad \Rightarrow \quad x > 11
\]

3. From \( 35 + x > 24 \):
\[
x > 24 - 35 \quad \Rightarrow \quad x > -11
\]
(This constraint doesn't affect our interval since \( x > 11 \) is the stronger condition.)

Combining the useful inequalities, we have:
\[
11 < x < 59
\]

So, the final answer is:

\[
11 < x < 59
\]

In the boxes:
- First box: 11
- Second box: 59