### Video Transcript

Is it possible to form a triangle with side lengths three inches, five inches, and seven inches?

Now, let’s recall the triangle inequality, which says the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. To have a better understanding of this rule, let’s look at a general triangle with vertices 𝐴, 𝐵, and 𝐶. According to the triangle inequality rule, if we take the length of two sides, such as side 𝐴𝐵 and 𝐵𝐶, the sum of those lengths must be greater than the length of the third side, which is 𝐴𝐶. This means that 𝐴𝐵 plus 𝐵𝐶 is greater than 𝐴𝐶.

So, that’s one formulation of the triangle inequality for this triangle. But it says the sum of the lengths of any two sides, which means we can also write this inequality down using other pairs of sides. So, it must also be true that 𝐴𝐵 plus 𝐴𝐶 is greater than 𝐵𝐶 and that 𝐵𝐶 plus 𝐴𝐶 is greater than 𝐴𝐵. So, whichever pair of sides I choose, the sum of those two side lengths must be greater than the length of the third side.

Now, we will return to our original question with lengths three, five, and seven. We want to find out if these three lengths form a triangle. So, we need to check if the triangle inequality holds true for all the different pairs of sides here. We’ve got three inequalities to check. Let’s be aware that even if one inequality fails, we cannot construct a triangle from those lengths. Let’s first check the sum of lengths three and five. We’re asking, is three plus five greater than seven? This inequality is of course true. However, we’re not done yet. We still need to check the other two inequalities.

Let’s take three and seven. Is three plus seven greater than five? Yes, that’s true as well. We’re not quite done until we check the third pair of side lengths. So, what’s left? We’ve already combined three with five and three with seven. We still have to combine five and seven. The final inequality of five plus seven greater than three is also true. We have demonstrated that all three triangle inequalities hold true with the lengths of three, five, and seven.

In conclusion, yes, it is possible to form a triangle with side lengths three inches, five inches, and seven inches. Our answer is backed up by the work that we showed, in which we checked that all three triangle inequalities held true.