Question

Chang knows one side of a triangle is 13 cm. Which set of two sides is possible for the lengths of the other two sides of this triangle?

5 cm and 8 cm
6 cm and 7 cm
7 cm and 2 cm
8 cm and 9 cm

Answer 1

To determine which set of two sides is possible for the lengths of the other two sides of a triangle when one side is known to be 13 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 third side.

Let \( a \), \( b \), and \( c \) be the lengths of the three sides of the triangle, where \( c \) is the longest side. In this case, one side (say \( c \)) is 13 cm, so we need to check each combination of two sides as follows:

1. **5 cm and 8 cm**
- 5 + 8 > 13: False (13 is not greater than 13)
- Thus, this combination does not form a triangle.

2. **6 cm and 7 cm**
- 6 + 7 > 13: False (13 is not greater than 13)
- Thus, this combination does not form a triangle.

3. **7 cm and 2 cm**
- 7 + 2 > 13: False (9 is not greater than 13)
- Thus, this combination does not form a triangle.

4. **8 cm and 9 cm**
- 8 + 9 > 13: True (17 is greater than 13)
- 8 + 13 > 9: True (21 is greater than 9)
- 9 + 13 > 8: True (22 is greater than 8)
- Thus, this combination does form a triangle.

Given these checks, the only set of two sides that can form a triangle with a side of 13 cm is **8 cm and 9 cm**.