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

5 cm
5 cm

10 cm
10 cm

6 cm
6 cm

8 cm
8 cm

Answer 1

To determine which of the given lengths cannot be a valid measure for the third side of a triangle with sides measuring 4 cm and 9 cm, we can apply 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's denote the sides of the triangle as \(a = 4\) cm, \(b = 9\) cm, and \(c\) as the third side.

Applying the triangle inequality, we have:

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

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

3. \(b + c > a\)
- \(9 + c > 4 \implies c > -5\) (this condition is always true as \(c\) must be positive)

From these inequalities, we can conclude:
- \(c < 13\) (From the first inequality)
- \(c > 5\) (From the second inequality)

Thus, for the third side \(c\), we must have:

\[
5 < c < 13
\]

Now, we will analyze each of the provided choices:

1. **5 cm**: This value is **not valid** since \(c\) must be greater than 5. So, it's not possible for \(c\) to be 5 cm.

2. **10 cm**: This value is valid since \(5 < 10 < 13\).

3. **6 cm**: This value is valid since \(5 < 6 < 13\).

4. **8 cm**: This value is valid since \(5 < 8 < 13\).

Therefore, the choice that **cannot be the measure of the third side** is:

**5 cm**.