Question Video: The Triangle Inequality Theorem Mathematics

Video Transcript

Use less than, equal to, or greater than to fill in the blank to compare side length 𝐵𝐶 to side length 𝐴𝐶.

First, we can identify which side lengths we’re trying to compare. We want to compare side length 𝐵𝐶 to side length 𝐴𝐶. One strategy we can try to use is the angle-side inequality in a triangle. To do that, we’d need to compare the angles opposite the two side lengths we’re interested in. Now you might be thinking we don’t have any information about the angles. However, we can use some properties of triangles to find out some information about the angle measures. In a triangle, if two sides have the same length, then their opposite angles are the same, which means here angle 𝐴𝐵𝐹 will be equal in measure to angle 𝐹𝐴𝐵 and angle 𝐴𝐷𝐹 will be equal to angle 𝐷𝐴𝐹.

Both triangle 𝐴𝐵𝐹 and triangle 𝐴𝐷𝐹 are isosceles triangles, and they have two equal sides. This means we can say that segment 𝐴𝐷 is going to be equal in length to segment 𝐴𝐵 as these two triangles are congruent. It also means angle 𝐴𝐹𝐵 will be equal to angle 𝐴𝐹𝐷. This means we found that side length 𝐴𝐵 must be smaller than side length 𝐴𝐶. But how can we say something about side length 𝐵𝐶? For this, we’re going to think about the line segment 𝐴𝐹. We’ve seen that the line segment 𝐴𝐹 bisects the segment 𝐵𝐷. The line segment 𝐴𝐹 is a bisector of the isosceles triangle 𝐴𝐵𝐷. And when that happens, it is a perpendicular bisector, which means both of these angles must measure 90 degrees. And if that’s the case, in the smaller isosceles triangles, the smaller blue angles must be equal to each other and must therefore be equal 45 degrees each.

If all of these smaller angles measure 45 degrees, the big angle we were looking for, angle 𝐵𝐴𝐶, is a right angle. And since line segment 𝐵𝐶 is opposite that right angle, it’s the hypotenuse of the larger triangle 𝐴𝐵𝐶, and it is therefore the longest side length of this triangle. The order of the side lengths for the larger triangle 𝐴𝐵𝐶 must be 𝐴𝐵 is smaller than 𝐴𝐶, which is smaller than 𝐵𝐶. Since 𝐵𝐶 is the hypotenuse and is the longest side length in this triangle, we can say 𝐵𝐶 is greater than 𝐴𝐶.