### Visualizing Laws About Triangles

Now I am going to show this in a diagram. And that will be much more interesting and very simple. So that is for your understanding. We can prove, this also in a diagram, just like how you visualize Pythagoras Theorem, in a very simple diagram… in a very interesting way. This BD2 =AD x DC. Let’s try visualizing. And that’s when you try visualizing – you can take 5 minutes – you try visualizing. And because if you can do that, that’s where you are thinking better and by thinking better 100% you are learning better and this is real learning.

Now we need to prove that BD2 =AD x DC. So replace BD by h, AD by x, and DC by y. So I need to prove h2 = xy. Shade the two inner triangles. The smaller triangle -rotate it by 90o like this, now complete the whole triangle. Un-shaded region is h2 as you can see here. Now keep this triangle here. Take a copy of this triangle. Now in the second one, interchange the two shaded triangles, like this. So that…and what is happening is – xh here, the order will become h followed by x. hy here – the order will become y,h. The total triangle – the length, the perpendicular sides – nothing is changing. Shaded regions are exactly the same – just that they are interchanged here. Now un-shaded region here is h2, un-shaded region here -it is xy, and they are equal. And it’s very simple and very interesting. So just like how we visualized Pythagoras Theorem- anything can be actually visualized- as you can see here it’s very simple…very simple and that’s why it’s so interesting. And once you learn like this, it’s very difficult to forget. Even if you want you won’t be able to forget. So I hope the complete section of similarity, similar triangles, and the theorems and the applications based on that is well below your level. And I am sure you will be able to do very well in exams. Because you are not just memorizing. Now you know what you are learning. You know why it’s working and you know where it can be used.