In triangle $$ A B D, \bar{A} B $$ is 3 cm long and $$ B D $$ is 5 cm long. According to the Triangle Inequality Theorem, which two lengths are possible lengths of $$ A \dot{D} $$ ? (1 point) $$ \square 4 \mathrm{~cm} $$ $$ \square 7 \mathrm{~cm} $$ $$ \square 2 \mathrm{~cm} $$ $$ \square 10 \mathrm{~cm} $$

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We are given that in triangle $$ABD$$: - $$ AB = 3 $$ cm - $$ BD = 5 $$ cm - Let $$ AD = x $$ cm. By the Triangle Inequality Theorem, the sum of the lengths of any two sides must be greater than the length of the third side. Therefore, we have three inequalities: 1. $$ AB + BD > AD $$ implies $$ 3 + 5 > x $$ or $$ x < 8 $$. 2. $$ AB + AD > BD $$ implies $$ 3 + x > 5 $$ or $$ x > 2 $$. 3. $$ AD + BD > AB $$ implies $$ x + 5 > 3 $$ which simplifies to $$ x > -2 $$ (this inequality is automatically satisfied since $$ x $$ is positive). Combining the first two nontrivial inequalities, we obtain: $$ 2 < x < 8. $$ Now, we check the given possible lengths: - $$ 4 $$ cm satisfies $$ 2 < 4 < 8 $$. - $$ 7 $$ cm satisfies $$ 2 < 7 < 8 $$. - $$ 2 $$ cm does not satisfy $$ x > 2 $$ (it is equal to 2, not greater than 2). - $$ 10 $$ cm does not satisfy $$ 10 < 8 $$. Thus, the possible lengths for $$ AD $$ are $$ 4 $$ cm and $$ 7 $$ cm. The possible lengths for $$ AD $$ are $$ 4 $$ cm and $$ 7 $$ cm.

In triangle is 3 cm long and is 5 cm long. According to the Triangle
Inequality Theorem, which two lengths are possible lengths of ? (1 point)

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In triangle is 3 cm long and is 5 cm long. According to the Triangle Inequality Theorem, which two lengths are possible lengths of ? (1 point)