To determine which of the choices cannot be the measure of the third side of a triangle when the other two sides measure 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 remaining side.
Let the lengths of the two known sides be \(a = 4\) cm and \(b = 9\) cm, and let \(c\) be the length of the third side. We need to check the following inequalities:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
Substituting the known values into the inequalities, we get:
- \(4 + 9 > c \Rightarrow 13 > c \Rightarrow c < 13\)
- \(4 + c > 9 \Rightarrow c > 5\)
- \(9 + c > 4 \Rightarrow c > -5\) (This condition is always satisfied since \(c\) must be a positive length.)
Combining the useful conditions, we find:
- \(c < 13\)
- \(c > 5\)
This means the possible range for the third side \(c\) is \(5 < c < 13\).
Now let's evaluate the given choices:
- 8 cm: \(5 < 8 < 13\) (valid)
- 6 cm: \(5 < 6 < 13\) (valid)
- 10 cm: \(5 < 10 < 13\) (valid)
- 5 cm: Not valid because \(c\) must be greater than 5.
Therefore, the measure of the third side that cannot be possible is 5 cm.