Question Video: Determining Whether a Triangle of Given Side Lengths Can Exist | Nagwa Question Video: Determining Whether a Triangle of Given Side Lengths Can Exist | Nagwa

Question Video: Determining Whether a Triangle of Given Side Lengths Can Exist Mathematics • Second Year of Preparatory School

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Does the triangle with side lengths 8, 32, and 16 exist?

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Video Transcript

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

To answer this question, let’s recall the triangle inequality. The triangle inequality helps us to determine whether given lengths can be sides of a triangle. And it states, “The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.” This means that if a triangle 𝐴𝐵𝐶 exists, then 𝐴𝐵 plus 𝐵𝐶 is greater than 𝐴𝐶, 𝐴𝐵 plus 𝐴𝐶 is greater than 𝐵𝐶, and 𝐵𝐶 plus 𝐴𝐶 is greater than 𝐴𝐵. 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 there’s no need to check the third inequality.

Because it was not possible to verify all three triangle inequalities were true, we can say, “No, a triangle with side lengths eight, 32, and 16 does not exist.”

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