### Class 6-10 Visualizing the Section Formula

Let me take the same point once again… you have this line AB… co-ordinates of A are (x1, y1)… co-ordinates of B are (x2, y2)… you have a point ‘P’ which has co-ordinates (x, y)… right? Now we know from this section formula that how you derive ‘x’ is (mx2 + nx1 / m + n) …right? Let’s see if we can visualize this…right? You have one straight line earlier when we understood section formula…we dropped perpendiculars and got these two triangles…right? These two triangles are similar…correct? So, if ‘AP : PB’ is ‘m : n’ …this will also be ‘m’ and this will be ‘n’…right? Similarly the third side, this will be ‘m’ and that will be ‘n’…right? Because these two are similar triangles…right? Now let’s draw some more lines and convert the triangles into rectangles…right? Let’s add some more triangles and convert this into rectangles…

Now this is the figure we have…let’s try to…now visualize what are section formula here was…so in section formula, you have a term mx2…how can you see mx2 in this diagram..? mx2 is nothing but area of this large rectangle…this large rectangle is nothing but mx2 …why? because this height is ‘m’ and this length is x2 …right? because the x co-ordinate of point ‘P’ is x2 …right? so this is ‘x2’, this is ‘m’…so this is nothing but the area mx2…ok? let’s shade this…we have this area mx2…now we need to add another value to it…we need to add nx1…right? Here, see you’re adding nx1… Can you visualize nx1 on this diagram..? can you visualize nx1 on this diagram? Area of this small rectangle is nothing but nx1…correct? Area of this small rectangle is nothing but nx1… because this is ‘n’…this distance is ‘n’ and this is x1 …correct? this is x1 which is nothing but the x co-ordinate of the point…so this rectangle is nx1… so now we will shade this also…right? because that is part of it…now you have this one large rectangle and this small rectangle…now look at these two rectangles…right? this side is ‘m’ and this side is ‘n’…similarly for this rectangle, this side is ‘m’ and this side is ‘n’…these two rectangles are actually one and the same…there’s no difference…right? these two rectangles are actually one and the same…right? so instead of having an odd-shaped figure…let’s just move this rectangle here…if I move this rectangle here…now I have a very simple rectangle…right? so this term that I had in my numerator (mx2 + nx1) is nothing but area of this rectangle…right? very simple…this term is nothing but area of this rectangle…what is this rectangle? this rectangle is nothing but a simple rectangle which has length ‘x’… length is ‘x’…right? …which the x co-ordinate of ‘P’…length is ‘x’ and breadth is (m + n)… correct? Length is ‘x’ and breadth is (m + n) … and the area I know is (mx2 + nx1) … now if I need to find out the length, what will I do..? I will divide the area by the breadth…right? I take m + n …. divide the area by it… I will get the length… that’s how you get…. ‘x’ is (mx2 + nx1 / m + n)…

This is the visualization… so now you understood section formula completely…right? just remembering that one formula does not mean knowing what section formula is…right? does not mean knowing what it is…does not mean you will be able to solve applications based questions…you will not be able to link it to physics…now you’ve truly understood the concept…right? because, now even without teaching you the moment of force in Physics, if I give you a problem you will be able to solve it…because it is nothing but section formula…right? similarly if you knew the concept in physics…right? how to support planks..? you will automatically know section formula…all of these are one and the same…right? and that’s how we need to learn the concept… now we truly understand this concept…right? we understand what the application is…right? which is planks with different weights and supporting them…right? Fulcrums and supports…we understand variable…what is the formula that actually represents this…how do you calculate the value of this support…we understand it diagrammatically…right? it is nothing but the area of this rectangle…right? what it is representing is nothing but area of this rectangle…and finally when we solve problems…what we are going to do is…use numbers…right? instead of these specific points…generic points… x1, y1 or x2, y2 …that we had…when we solve a problem, we’ll have specific values for this…right? each of these x1, y1, x2, y2 … all of them will have specific numbers…right? so once we solve problems where we use specific numbers, we would have completed the concept in all possible ways…right? now you completely understand section formula, you don’t need to mug-up anything…right? you truly understand the concept…let’s see if we can now solve a problem…