Question

Two sides of a triangle measure 4 cm and 9 cm. Determine which of the following choices cannot be the measure of the third side.(1 point)
Responses

6 cm
6 cm

10 cm
10 cm

8 cm
8 cm

5 cm
5 cm

Answer 1

To determine which of the given lengths cannot be the measure of the third side of a triangle when the other two sides are 4 cm and 9 cm, we can apply 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.

Let \( a = 4 \) cm and \( b = 9 \) cm, and let \( c \) be the length of the third side. The triangle inequalities can be expressed as follows:

1. \( a + b > c \)
- \( 4 + 9 > c \)
- \( 13 > c \)
- Thus, \( c < 13 \)

2. \( a + c > b \)
- \( 4 + c > 9 \)
- \( c > 5 \)

3. \( b + c > a \)
- \( 9 + c > 4 \)
- This inequality is always true since \( c \) is positive.

From the inequalities, we derive that \( c \) must satisfy:
- \( 5 < c < 13 \)

Now, let's check the provided options:
- **6 cm:** This satisfies \( 5 < 6 < 13 \) (valid)
- **10 cm:** This satisfies \( 5 < 10 < 13 \) (valid)
- **8 cm:** This satisfies \( 5 < 8 < 13 \) (valid)
- **5 cm:** This does not satisfy \( c > 5 \) (invalid)

Therefore, the measure that cannot be the length of the third side is **5 cm**.