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

10 cm
10 cm

6 cm
6 cm

5 cm
5 cm

8 cm

Answer 1

To determine the possible measures for the third side of a triangle given two sides of lengths 4 cm and 9 cm, we can use the triangle inequality theorem. This theorem states that for any triangle with sides of lengths \( a \), \( b \), and \( c \):

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

Let's denote the lengths of the two known sides as \( a = 4 \) cm and \( b = 9 \) cm, and the length of the third side as \( c \).

Applying the triangle inequality:

1. \( 4 + 9 > c \) → \( 13 > c \) → \( c < 13 \)
2. \( 4 + c > 9 \) → \( c > 5 \)
3. \( 9 + c > 4 \) → \( c > -5 \) (this inequality is always true since lengths can’t be negative)

From the inequalities we derived, the possible range for the length of the third side \( c \) is:
\[ 5 < c < 13 \]

Now let's evaluate the possible choices:
- **10 cm**: This value is within the range (5 < 10 < 13).
- **6 cm**: This value is also within the range (5 < 6 < 13).
- **5 cm**: This value is **not** within the range (since \( c \) must be greater than 5).
- **8 cm**: This value is within the range (5 < 8 < 13).

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