Question

A triangle has side lengths of 11 cm and 7 cm. Which could be the value of the third side, 19 cm or 17 cm?

Answer 1

To determine whether a third side of a triangle can measure 19 cm or 17 cm, we can use the triangle inequality theorem. The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let's denote the sides of the triangle as follows:
- Side a = 11 cm
- Side b = 7 cm
- Side c = third side (either 19 cm or 17 cm)

We need to check both potential values of the third side.

1. **Checking c = 19 cm**:
- Check \( a + b > c \):
\( 11 + 7 > 19 \)
\( 18 > 19 \) (False)
- Check \( a + c > b \):
\( 11 + 19 > 7 \)
\( 30 > 7 \) (True)
- Check \( b + c > a \):
\( 7 + 19 > 11 \)
\( 26 > 11 \) (True)

Since one of the inequalities is false (specifically \( 11 + 7 > 19 \)), a triangle with sides 11 cm, 7 cm, and 19 cm cannot exist.

2. **Checking c = 17 cm**:
- Check \( a + b > c \):
\( 11 + 7 > 17 \)
\( 18 > 17 \) (True)
- Check \( a + c > b \):
\( 11 + 17 > 7 \)
\( 28 > 7 \) (True)
- Check \( b + c > a \):
\( 7 + 17 > 11 \)
\( 24 > 11 \) (True)

Since all inequalities are satisfied, a triangle with sides 11 cm, 7 cm, and 17 cm can exist.

**Conclusion**: The value of the third side could be 17 cm, but it cannot be 19 cm.