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?

A. 19 cm
B. 17 cm

Answer 1

To determine which of the given lengths could be the length of the third side of the triangle, 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 third side.

Given the side lengths of 11 cm and 7 cm, we can evaluate the potential third sides:

1. If the third side is \( c \), then the triangle inequalities that must be satisfied are:

- \( 11 + 7 > c \)
- \( 11 + c > 7 \)
- \( 7 + c > 11 \)

Let's analyze these inequalities for each proposed value for \( c \).

### Testing 19 cm:
1. \( 11 + 7 > 19 \) → \( 18 > 19 \) (False)
2. \( 11 + 19 > 7 \) → \( 30 > 7 \) (True)
3. \( 7 + 19 > 11 \) → \( 26 > 11 \) (True)

Since the first inequality is false, a side length of 19 cm cannot form a triangle with the given sides.

### Testing 17 cm:
1. \( 11 + 7 > 17 \) → \( 18 > 17 \) (True)
2. \( 11 + 17 > 7 \) → \( 28 > 7 \) (True)
3. \( 7 + 17 > 11 \) → \( 24 > 11 \) (True)

All the inequalities are satisfied for a side length of 17 cm.

### Conclusion:
The only value that could be the length of the third side of the triangle is:

**B. 17 cm**.