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jmc

geometry senior

Problem

The sides of a triangle with positive area have lengths 4, 6, and . The sides of a second triangle with positive area have lengths 4, 6, and . What is the smallest positive number that is a possible value of ?
Solution
By the Triangle Inequality, each of and can be any number strictly between 2 and 10, so . Therefore, the smallest positive number that is not a possible value of is .
Final answer
8