### Video Transcript

Does the triangle with side lengths eight, 32, and 16 exist?

To answer this question, letโs recall the triangle inequality. The triangle inequality helps us to determine whether given lengths can be sides of a triangle. And it states, โThe sum of the lengths of any two sides of a triangle must be greater than the length of the third side.โ This means that if a triangle ๐ด๐ต๐ถ exists, then ๐ด๐ต plus ๐ต๐ถ is greater than ๐ด๐ถ, ๐ด๐ต plus ๐ด๐ถ is greater than ๐ต๐ถ, and ๐ต๐ถ plus ๐ด๐ถ is greater than ๐ด๐ต. For a triangle with side lengths eight, 32, and 16 to exist, those numbers must make all three triangle inequalities true.

First, we will ask ourselves if eight plus 32 is greater than 16, and that is true. Moving on to the second inequality, we ask ourselves if eight plus 16 is greater than 32. This statement is false because 24 is not greater than 32. As soon as we find one of the triangle inequalities is not true, we know it is not possible to form a triangle. So thereโs no need to check the third inequality.

Because it was not possible to verify all three triangle inequalities were true, we can say, โNo, a triangle with side lengths eight, 32, and 16 does not exist.โ