@bravekiwi Yes, you are right that at the very end, Prof. Loh says, "This question shows us what happens when you take any old 30-60-90 right triangle and draw a line though the center." Even though the mini-question at the end of the Your Turn (Part 2) video seems like a very different triangle, it's actually illustrating the same concept as in the original Your Turn question! We could even use a random triangle to illustrate this concept, which has a nice name: it's called the Angle Bisector Theorem.
When you have a line bisecting an angle in a triangle, that line will cut the opposite side in two parts, the ratios of which are in the same ratio as the other two sides of the triangle!
So the ratio of the green and purple sides, \(13:12,\) is the same as the ratio of the segments cut from the third side, \(13:12.\)
In fact, if we had used another \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, then it might not have been obvious that this fact works for any triangle! It might have appeared to be a property of only \(30^{\circ}-60^{\circ}-90^{\circ}\) right triangles.