Question Video: The Triangle Inequality Theorem Mathematics

Use <, =, or > to fill in the blank: 𝐡𝐢 _ 𝐴𝐢.

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Video Transcript

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

First, we can identify which side lengths we’re trying to compare. We want to compare side length 𝐡𝐢 to side length 𝐴𝐢. 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 𝐴𝐡𝐹 will be equal in measure to angle 𝐹𝐴𝐡 and angle 𝐴𝐷𝐹 will be equal to angle 𝐷𝐴𝐹.

Both triangle 𝐴𝐡𝐹 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 𝐴𝐡 as these two triangles are congruent. It also means angle 𝐴𝐹𝐡 will be equal to angle 𝐴𝐹𝐷. This means we found that side length 𝐴𝐡 must be smaller than side length 𝐴𝐢. But how can we say something about side length 𝐡𝐢? For this, we’re going to think about the line segment 𝐴𝐹. We’ve seen that the line segment 𝐴𝐹 bisects the segment 𝐡𝐷. The line segment 𝐴𝐹 is a bisector of the isosceles triangle 𝐴𝐡𝐷. 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 𝐡𝐴𝐢, is a right angle. And since line segment 𝐡𝐢 is opposite that right angle, it’s the hypotenuse of the larger triangle 𝐴𝐡𝐢, and it is therefore the longest side length of this triangle. The order of the side lengths for the larger triangle 𝐴𝐡𝐢 must be 𝐴𝐡 is smaller than 𝐴𝐢, which is smaller than 𝐡𝐢. Since 𝐡𝐢 is the hypotenuse and is the longest side length in this triangle, we can say 𝐡𝐢 is greater than 𝐴𝐢.

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