Question Video: Applying the Triangle Inequality to a Right Triangle Mathematics

Video Transcript

In a triangle 𝐴𝐵𝐶, if the measure of angle 𝐵 is equal to 90 degrees, which of the following is true? Option (A) 𝐴𝐵 is greater than 𝐴𝐶. Option (B) 𝐴𝐵 is equal to 𝐴𝐶. Option (C) 𝐴𝐶 is greater than 𝐴𝐵. Or is it option (D) 𝐵𝐶 is greater than 𝐴𝐶?

In this question, we are given that the measure of one of the angles in a triangle is 90 degrees. And we need to use this information to determine the correct statement comparing the side lengths in the triangle.

To answer this question, let’s start by sketching the information we have been given. We have a triangle 𝐴𝐵𝐶 with a right angle at 𝐵 as shown. We can then note that since 𝐴𝐵𝐶 is a right triangle, the side opposite the right angle will be the hypotenuse, that is, side 𝐴𝐶. We know that the hypotenuse is the longest side in a right triangle. We can see that this matches option (C), since this states that the length of 𝐴𝐶 is greater than the length of 𝐴𝐵.

It is also worth noting that none of the other options can be correct, since side 𝐴𝐶 has greater length than all of the other sides, since it is the hypotenuse.

It is also possible to answer this question without using the properties of right triangles. To do this, we will use the side comparison theorem in triangles. First, we know that the sum of the measures of the internal angles in a triangle is 180 degrees. So the measure of angle 𝐴 plus the measure of angle 𝐵 plus the measure of angle 𝐶 is 180 degrees. Next, we know that the measure of angle 𝐵 is 90 degrees. So we can substitute this into the equation and subtract 90 degrees from both sides to obtain that the measure of angle 𝐴 plus the measure of angle 𝐶 is equal to 90 degrees.

Now we can note that since we have a triangle, the measures of the angles 𝐴 and 𝐶 must be positive. And for the sum of two positive numbers to be equal to 90, both of these numbers must be less than 90. So the angle at 𝐵 is the angle with the largest measure in the triangle.

We can now apply the side comparison theorem in triangles that states that the side opposite the angle with larger measure in a triangle will be longer. Since we have shown that the measure of angle 𝐵 is greater than the measure of the other two angles, the side opposite this angle must be longer than the other two sides. So 𝐴𝐶 is greater than 𝐵𝐶 and 𝐴𝐶 is greater than 𝐴𝐵.

This is a much longer method to show that the answer is option (C). However, there is one useful property that we can obtain from this method. This exact method will work on any triangle with an angle of measure greater than or equal to 90 degrees. We can show that this angle will have the largest measure in the triangle, so the side opposite this angle is the longest.

In either case, we have shown that in a triangle 𝐴𝐵𝐶, where the measure of angle 𝐵 is 90 degrees, we must have that 𝐴𝐶 is greater than 𝐴𝐵, which is option (C).