A triangle has two sides of lengths 5 and 8. What is the range of possible lengths for the third side length, which is represented by x? Type the numerical values in the boxes and remember to put them in the correct order.(2 points)

Answer 1

To determine the range of possible lengths for the third side \( x \) of a triangle with side lengths of 5 and 8, we can use the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

We have three inequalities to consider:

1. \( 5 + 8 > x \)
2. \( 5 + x > 8 \)
3. \( 8 + x > 5 \)

Now, let's solve each inequality one by one.

1. From \( 5 + 8 > x \): \[ x < 13 \]

2. From \( 5 + x > 8 \): \[ x > 3 \]

3. From \( 8 + x > 5 \): \[ x > -3 \quad (\text{This is always true if } x > 3) \]

Combining the valid inequalities, we find: \[ 3 < x < 13 \]

Thus, the range of possible lengths for \( x \) is:

• Lower bound: 3
• Upper bound: 13

So the final answer is:

3 and 13.