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.
- 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 \(b\):
- \(4 + 9 > c \Rightarrow 13 > c\) (or \(c < 13\))
- \(4 + c > 9 \Rightarrow c > 5\)
- \(9 + c > 4 \Rightarrow c > -5\) (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.