Grades 11-12 Video Solutions 2012-2012_11-12

Video Transcription

What is the smallest possible size of an angle in an isosceles triangle ABC that has a median that divides the triangle into two isosceles triangles?  So here's an isosceles triangle. We've drawn in one of the medians. Later on, we're going to do the case when the median comes from this top.  But let's just do this case first. So these two sides are equal because that's what medians do.  We also know that these two angles are equal right here because this triangle is isosceles.  Okay, so let's take a look at this triangle down here with these three angles.  So this green angle is smaller than this total angle right here, which means that it must be smaller than this blue angle. So that means these two are not equal.  We also know that this triangle is isosceles, so it has two equal angles. That means this red angle has to either be equal to the green angle or to the blue angle.  Okay, so since this triangle is isosceles and this red angle is a base angle, it has to be less than 90 degrees. I mean, imagine if it wasn't,  then you'd have another angle that's equal to it, and that just makes it have too many degrees in this triangle.  So since this angle is less than 90 degrees, this angle has to be greater than 90 degrees. So this triangle right here is isosceles, and here's an angle greater than 90 degrees.  That means these two angles have to be equal to each other,  which tells us that this side is equal to this side.  But then that tells us that the green angle and the blue angle are equal, which is absurd because the green angle is smaller than the blue angle.  Okay, so that means this case is just gone.  So now let's try this case.  We want this triangle to be isosceles and this triangle to be isosceles, but this is a median, and it comes from, like, this top angle. So we know that it's also an altitude,  meaning this is a right angle right here. So these are both right isosceles triangles. So let's redraw that to be the case, and there we go.  That's just the only way that it can happen. Now, that means the smallest angle in our original triangle is right here, 45 degrees,  and so that has to be our answer.

Video Summary

The smallest possible angle in an isosceles triangle with a median that divides it into two isosceles triangles is 45 degrees. Initially, it was shown that if the median does not originate from the vertex of the apex angle, this leads to contradictions involving angle sizes, implying that no valid configuration exists. In the corrected setup, where the median acts as an altitude from the apex, each resulting triangle is a right isosceles triangle, establishing that the smallest angle possible in the original configuration is 45 degrees.

Keywords

isosceles triangle

median

smallest angle

right isosceles triangle

45 degrees