Question Video: Recalling the Triangle Inequality | Nagwa Question Video: Recalling the Triangle Inequality | Nagwa

Question Video: Recalling the Triangle Inequality Mathematics • Second Year of Preparatory School

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Complete the sentence. The sum of the lengths of any two sides in a triangle is ＿ the length of the other side.

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Video Transcript

Complete the sentence. The sum of the lengths of any two sides in a triangle is what the length of the other side. Option (A) equal to. Option (B) greater than. Or is it option (C) less than?

In this question, we are asked to compare the sum of the lengths of two sides in a triangle with the length of its other side by filling in the blank in a sentence. The easiest way to answer this question is to recall that the triangle inequality tells us information about the comparison of the sum of the side lengths of two sides in a triangle with the length of the third side. In particular, we can recall that the triangle inequality tells us that the sum of any two side lengths in a triangle must be greater than the length of the third side. This means that the answer is option (B). The sum of the lengths of any two sides in a triangle is greater than the length of third side by the triangle inequality.

Although this is enough to answer this question, it is always worth understanding why this is the answer. To do this, let’s ask a slightly different question. What is the shortest distance between any two points? Let’s say we want to travel between these two points, 𝑃 and 𝑄. Intuitively, the shortest distance between the points must be the straight line between them. If we wanted to travel somewhere quickly, we would want to travel there directly instead of taking a detour.

We can extend this idea even further. If the shortest distance between 𝑃 and 𝑄 is a straight line, then traveling via any other point 𝑅 not on this line must be a longer distance. If we call the lengths of the sides of the triangle 𝑎, 𝑏, and 𝑐 as shown, then we have argued that 𝑎 plus 𝑏 must be greater than 𝑐, since 𝑐 is the length of the shortest distance between the points and 𝑎 plus 𝑏 must be a longer distance.

It is worth noting that this is not a proof of the triangle inequality, since we have made several assumptions which we have not proven. However, it is a useful intuitive reason why this inequality holds true. In either case, we have shown that the sum of the lengths of two sides in a triangle is greater than the length of the other side by using the triangle inequality.

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