To determine the possible measures for the third side of a triangle given two sides of lengths 4 cm and 9 cm, we can use the triangle inequality theorem. This theorem states that for any triangle with sides of lengths \( a \), \( b \), and \( c \):
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Let's denote the lengths of the two known sides as \( a = 4 \) cm and \( b = 9 \) cm, and the length of the third side as \( c \).
Applying the triangle inequality:
- \( 4 + 9 > c \) → \( 13 > c \) → \( c < 13 \)
- \( 4 + c > 9 \) → \( c > 5 \)
- \( 9 + c > 4 \) → \( c > -5 \) (this inequality is always true since lengths can’t be negative)
From the inequalities we derived, the possible range for the length of the third side \( c \) is: \[ 5 < c < 13 \]
Now let's evaluate the possible choices:
- 10 cm: This value is within the range (5 < 10 < 13).
- 6 cm: This value is also within the range (5 < 6 < 13).
- 5 cm: This value is not within the range (since \( c \) must be greater than 5).
- 8 cm: This value is within the range (5 < 8 < 13).
Thus, the measure of the third side that cannot be the correct length is 5 cm.
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