Question Video: Applying the Triangle Inequality Theorem to a Figure Mathematics

Video Transcript

Use is less than, is equal to, or is greater than to fill in the blank. 𝑀𝐴 plus 𝑀𝐵 plus 𝑀𝐶 what one-half 𝐴𝐵 plus 𝐴𝐶 plus 𝐵𝐶.

In this question, we are asked to compare the sizes of two expressions which involve the sum of side lengths in triangles. So, we will start by recalling the triangle inequality. We recall that this tells us that in any triangle the sum of the lengths of any two sides in the triangle must be greater than the length of the remaining side. We can use the triangle inequality and the figure to construct inequalities involving the lengths of sides in triangles.

Let’s start by applying the triangle inequality to triangle 𝐴𝐵𝑀. We know that the sum of the lengths of any two sides in this triangle must be greater than the length of the remaining side. So, 𝑀𝐴 plus 𝑀𝐵 is greater than 𝐴𝐵. We can apply this result once more, this time to triangle 𝐵𝐶𝑀, to get that 𝑀𝐵 plus 𝑀𝐶 is greater than 𝐵𝐶. We can then apply the triangle inequality one final time, this time to triangle 𝐴𝐶𝑀, to obtain that 𝑀𝐴 plus 𝑀𝐶 is greater than 𝐴𝐶.

We now have three inequalities involving the sum of the side lengths in the question. We want to add these inequalities together to find an inequality linking the sums of side lengths in the question.

Adding the left-hand sides of the three inequalities together, we can see that we have two times each side length. So, the left-hand side of the sum of these three inequalities is two 𝑀𝐴 plus two 𝑀𝐵 plus two 𝑀𝐶. We can then add the side lengths on the right-hand sides of the inequalities to get 𝐴𝐵 plus 𝐴𝐶 plus 𝐵𝐶. Finally, we multiply both sides of this inequality by one-half. This gives us that 𝑀𝐴 plus 𝑀𝐵 plus 𝑀𝐶 is greater than one-half 𝐴𝐵 plus 𝐴𝐶 plus 𝐵𝐶. So, the answer is greater than.