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.
