Two sides of a triangle have lengths of 5 and 19. Can the third side have a length of 13?

Answer

To determine if a triangle can have sides of lengths 5, 19, and 13, we will use the triangle inequality theorem. This theorem states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following conditions must be satisfied:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

Let's denote the sides as follows:
- \(a = 5\)
- \(b = 19\)
- \(c = 13\)

Now, we will check the triangle inequality conditions step by step.

### Step 1: Check \(a + b > c\)
Calculate:
\[ 5 + 19 = 24 \]
Now check:
\[ 24 > 13 \quad \text{(True)} \]

### Step 2: Check \(a + c > b\)
Calculate:
\[ 5 + 13 = 18 \]
Now check:
\[ 18 > 19 \quad \text{(False)} \]

### Step 3: Check \(b + c > a\)
Calculate:
\[ 19 + 13 = 32 \]
Now check:
\[ 32 > 5 \quad \text{(True)} \]

### Conclusion
Since the second condition \(a + c > b\) is false, the lengths 5, 19, and 13 do not satisfy the triangle inequality theorem. Therefore, a triangle cannot be formed with these side lengths.

### Final Answer
The third side of the triangle cannot have a length of 13.

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