To determine which of the given lengths cannot be a valid measure for the third side of a triangle with sides measuring 4 cm and 9 cm, we can apply 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 \(a = 4\) cm, \(b = 9\) cm, and \(c\) as the third side.
Applying the triangle inequality, we have:
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\(a + b > c\)
- \(4 + 9 > c \implies 13 > c \implies c < 13\)
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\(a + c > b\)
- \(4 + c > 9 \implies c > 5\)
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\(b + c > a\)
- \(9 + c > 4 \implies c > -5\) (this condition is always true as \(c\) must be positive)
From these inequalities, we can conclude:
- \(c < 13\) (From the first inequality)
- \(c > 5\) (From the second inequality)
Thus, for the third side \(c\), we must have:
\[ 5 < c < 13 \]
Now, we will analyze each of the provided choices:
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5 cm: This value is not valid since \(c\) must be greater than 5. So, it's not possible for \(c\) to be 5 cm.
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10 cm: This value is valid since \(5 < 10 < 13\).
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6 cm: This value is valid since \(5 < 6 < 13\).
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8 cm: This value is valid since \(5 < 8 < 13\).
Therefore, the choice that cannot be the measure of the third side is:
5 cm.
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