Find the range of possible measures of the third side. $$ 8,6 $$ 16,19

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To find the range of possible measures for the third side of a triangle when the lengths of the other two sides are given, we can use the triangle inequality theorem. The 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. Given the two sides $$ a = 8 $$ and $$ b = 6 $$, we can denote the third side as $$ c $$. The triangle inequalities can be expressed as: 1. $$ a + b > c $$ 2. $$ a + c > b $$ 3. $$ b + c > a $$ Let's apply these inequalities step by step. ### Step 1: Apply the inequalities 1. From $$ a + b > c $$: $$ 8 + 6 > c \implies 14 > c \implies c < 14 $$ 2. From $$ a + c > b $$: $$ 8 + c > 6 \implies c > 6 - 8 \implies c > -2 $$ (This inequality is not restrictive since $$ c $$ must be positive.) 3. From $$ b + c > a $$: $$ 6 + c > 8 \implies c > 8 - 6 \implies c > 2 $$ ### Step 2: Combine the results From the inequalities, we have: - $$ c < 14 $$ - $$ c > 2 $$ Thus, the range of possible measures for the third side $$ c $$ is: $$ 2 < c < 14 $$ ### Conclusion The range of possible measures for the third side is $$ (2, 14) $$. The third side must be greater than 2 and less than 14.

Find the range of possible measures of the third
side.

16,19

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Solution

Step 1: Apply the inequalities

Step 2: Combine the results

Conclusion

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Find the range of possible measures of the third side. 16,19