To apply the Triangle Inequality Theorem, we need to ensure 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 first convert the measurements to the same unit.
The first side is given as 6 1/2 cm, which can be converted to an improper fraction: \[ 6 \frac{1}{2} \text{ cm} = \frac{13}{2} \text{ cm} \approx 6.5 \text{ cm} \] The second side is 10 mm, which we can convert to centimeters (since there are 10 mm in 1 cm): \[ 10 \text{ mm} = 1 \text{ cm} \]
Now, we have the two sides of the triangle:
- Side 1 = 6.5 cm
- Side 2 = 1 cm
Let \( x \) be the length of the third side. According to the Triangle Inequality Theorem, we have the following inequalities:
- \( 6.5 + 1 > x \)
- \( 6.5 + x > 1 \)
- \( 1 + x > 6.5 \)
Let's solve these inequalities one by one:
-
\( 6.5 + 1 > x \) \[ 7.5 > x \] (or \( x < 7.5 \))
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\( 6.5 + x > 1 \) \[ x > 1 - 6.5 \] \[ x > -5.5 \] (This condition will always be true since the side length must be positive)
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\( 1 + x > 6.5 \) \[ x > 6.5 - 1 \] \[ x > 5.5 \]
Putting it all together, \( x \) must satisfy: \[ 5.5 < x < 7.5 \]
Now, reviewing the possible responses:
- 6 cm (fits since \( 5.5 < 6 < 7.5 \))
- 16 cm (does not fit)
- 18 cm (does not fit)
- 8 cm (does not fit)
So, the possible measure of the third side is 6 cm.
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