Using Triangle Inequalities

We asked to provide a proof for the following problem in our last newsletter. We are publishing the best explained response. Lookout for our future newsletter. Your response can also be published. It’s important to continue to learn by proofs.

Can you prove it?

A straight bar of length 2 m is cut into five pieces with each piece at least 17 cm long. Prove that there are three of these pieces that can be put together to form a triangle.

Proof BY Ansh Agarwal

We can start by reviewing a key concept which is called triangular inequalities. According to this concept we can not make a triangle if sum of the lengths of any two sides is less than or equal to the third side.

Let’s assume that we can cut the 2 meter bar, such that we can’t make a triangle with the pieces. We want to minimize the length we use, so we can let the lengths of the first 2 bars be a = b = 17 cm each as 17 is the length of the smallest piece. Letting other three sides be c, d, and e, from the triangle inequality we can choose minimum lengths of the pieces so that no triangles can be formed:

Hence, the total length of all pieces will be 17 + 17 + 34 + 51 + 85 = 204 cm. This is bigger than 2 meter. So, our assumption that we can cut the bar such that we can’t form a triangle is false, and so, no matter how we cut the bar, there are 3 pieces that can make a triangle.

To confirm, we can reduce size of c, d, or e to fit all pieces in 2 meter size. This way we are guaranteed to make a triangle using the pieces, e.g. if we reduce e by 4 cm to make overall length of all pieces equal to 2 meter, it means e = 81 cm then pieces c, d, and e will make a triangle with side length of 34, 51, 81 because 81<34+51. You may try other combinations as well.

Thanks Ansh Agarwal for sending us a detailed proof. You can learn more about Ansh here. You can also add your comments if you have additional thoughts on proving this.