To determine which side lengths cannot form a triangle, we can use 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 the lengths of the two given sides be \( a = 4 \) cm and \( b = 9 \) cm, and let \( c \) be the length of the third side. The triangle inequalities we need to satisfy are:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Substituting the known values:
- \( 4 + 9 > c \) ⇒ \( 13 > c \) ⇒ \( c < 13 \)
- \( 4 + c > 9 \) ⇒ \( c > 5 \)
- \( 9 + c > 4 \) ⇒ This inequality is always true since \( c \) will be positive.
Thus, we have the constraints:
- \( c < 13 \)
- \( c > 5 \)
From these inequalities, the possible values for \( c \) must be in the range \( 5 < c < 13 \).
Now, let's look at the provided options: 6 cm, 10 cm, 8 cm, 5 cm.
- 6 cm: \( 5 < 6 < 13 \) (Valid)
- 10 cm: \( 5 < 10 < 13 \) (Valid)
- 8 cm: \( 5 < 8 < 13 \) (Valid)
- 5 cm: \( 5 \not< 5 < 13 \) (Not valid)
The length that cannot be the measure of the third side is 5 cm.