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

5 cm
5 cm

6 cm
6 cm

8 cm
8 cm

Answer 1

hello

Answer 2

To determine which side length cannot be the measure of the third side of a triangle with two sides measuring 4 cm and 9 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's denote the sides of the triangle as \(a = 4 \, \text{cm}\), \(b = 9 \, \text{cm}\), and \(c\) as the third side.

1. According to the triangle inequality, we need to check the following conditions:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)

Now substituting the values for \(a\) and \(b\):

1. \(4 + 9 > c \Rightarrow 13 > c\) (or \(c < 13\))
2. \(4 + c > 9 \Rightarrow c > 5\)
3. \(9 + c > 4 \Rightarrow c > -5\) (this condition is always satisfied since \(c\) must be positive)

From these inequalities, we conclude:
- \(c < 13\)
- \(c > 5\)

This means the possible values for \(c\) must be in the range \(5 < c < 13\).

Now let's check each option:
- 10 cm: \(5 < 10 < 13\) (valid)
- 5 cm: \(5\) is not greater than \(5\) (not valid)
- 6 cm: \(5 < 6 < 13\) (valid)
- 8 cm: \(5 < 8 < 13\) (valid)

Thus, the measure of the third side that **cannot** be valid is:
**5 cm**.