To determine which length cannot be the measure of the third side of a triangle when the other two sides are 4 cm and 9 cm, we can use the triangle inequality theorem. The triangle inequality states that for any triangle with sides of lengths \( a \), \( b \), and \( c \):
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Let the sides be:
- \( a = 4 \) cm
- \( b = 9 \) cm
- \( c \) is the length of the third side.
Now apply the triangle inequality:
-
\( 4 + 9 > c \)
- This simplifies to \( c < 13 \).
-
\( 4 + c > 9 \)
- This simplifies to \( c > 5 \).
-
\( 9 + c > 4 \)
- This simplifies to \( c > -5 \) (always true for positive lengths).
So the conditions we have are:
- \( c < 13 \)
- \( c > 5 \)
This means the possible values for the third side \( c \) must satisfy: \[ 5 < c < 13 \]
Now let's examine the provided options:
- 10 cm: Valid (since \( 5 < 10 < 13 \))
- 6 cm: Valid (since \( 5 < 6 < 13 \))
- 5 cm: Not Valid (since \( 5 \) is not greater than \( 5 \))
- 8 cm: Valid (since \( 5 < 8 < 13 \))
Thus, the third side cannot be 5 cm.