 Hello friends welcome again to another session on elements of Indian mathematics. In this session we are going to do a very interesting construction. Now this construction has been discussed by Euclid as well, but we are going to take a look at how Indian mathematicians and precisely how the scholars who wrote Sulba Sutras have you know discussed this construction or have done this construction. What is this construction all about? So basically we are going to discuss a method of constructing a square equal in area that of a rectangle once again. So we are going to construct a square equal in area of that of given rectangle. So let us begin the construction. So for that I am going to first draw a rectangle. So let us say I am drawing this rectangle. So let me draw the rectangle first. So let me draw the polygon rectangle. So you can see I am drawing this A, B, C, D. Perfect. So this is a rectangle, the area, how much is the area? Let us see its area. So its area is 54 okay. So let us just hide this area for the time being. Now what we are going to do is we are going to construct a square whose area is 54. So how do I do it? So first I will do the construction. So pay attention to the steps which I am following and then I am going to prove that also why it is working okay. So let us first draw or complete the construction. So what I am going to do first is I am going to draw a square with length DC here or cut out a square with length or side DC. So how do I do it? So basically I will take a circle and this circle is centered at C and it passes through D. So clearly this point E, so C E is equal to C D right, the radii of the same circle so done. Now what I am going to do is I am going to draw a perpendicular line on E perpendicular to this line yeah. So if you look at this point F right, so D C E F is the required square okay. Now I am going to hide these things because I do not need them anymore. So let me hide these okay. So and let me join F E clear. So F E is a square whose side is the breadth of the given rectangle this is the first step. Now what I am going to do is I am going to divide this rectangle into two halves, two halves, equal halves. So for that I need to take the midpoint. So let me take the midpoint of A and F. So G is the midpoint of A F and now I am going to draw another perpendicular line on G on A F like that okay. So if you look at it this these two rectangles so A G H B and G F E H R equal in area is it. So now let me again take this away I do not require it and let me draw the segment G H done. So what have we done so far? We have created rectangle random rectangle whose area was 54 square units. Then I constructed a square with the base same as the or the side same as that of the breadth of the rectangle. Then whatever is the leftover portion of the rectangle I divided into two parts okay this one and this one. So what I am going to do is I am going to reproduce this rectangle on the sides here EC what do I mean is I have to create a rectangle like that whose area is same as any of these rectangles. So let me do that how do I do it? So let me have that line back which line or rather let me draw a line this line F E connected line right. So this is the line. So what I am going to do is I am going to draw a circle with center E and radius E H. Now you can see this E and let me name this point this point is I. So E I is equal in length to E H radii of the same circle okay. And now if I drop a perpendicular from I so let me name this point as J so what do you think guys I J length is equal to CE is it rectangle perfect rectangle and E I is same as E H. So in a way this rectangle look at this rectangle E C J and I is equal in area to G F E H is it why length is same length is same so F E is the length here EC is the length here and E I is the breadth here and H is the breadth here so both are same. So hence the area of again E C J I is equal to H G F E right is that okay. So I hope this is clear so let me now take away the circle I don't require it I don't require it either I don't require it anymore. So let me now construct the rectangle one for all let me complete this rectangle so E I and I J and J C okay guys so EC J I is equal in area to G G F E H or A G H B fine now what I am going to do is I am going to first of all let me also extend these lines so there was few lines let me open them up yeah so I need I will be leading this line yeah and I will also be needing this line okay perfect now. So look carefully what I am going to do now I taking J and this point which point here K J K J K as the radius and J as the center so look J as the center and K as the radius I have now found out a point where the circle is intersecting this H E. So this point is let me make it a little bit more zoomed in so yeah H L I hope this is visible to all of you so let me put this L here so these two points I hope this is clear right let me take it to the center here yeah okay perfect so you can see this H L right now another thing which I need to do is I need to drop a perpendicular from this point L sorry this point L and on to this line okay perfect now when you do this this line here the line perpendicular to H E on L cuts I K here let me zoom in again and yeah M so M is the right now now what am I going to do I am going to draw a square with side M J M J once again M J okay and this M J is the required side of the square so let me draw a circle like that okay and so this is one point so let me name this point here and and again a circle from N or rather I need not draw a circle I can just drop a perpendicular line from N on to this so here is the point of intersection where is the point of intersection this one oh and let me draw a polygon again with M J N O back to back to M no sorry the last one was not done properly so once again so let me zoom in first for a clear view okay so let me do this now am I what am I doing I am joining the I am creating this polygon M J N O and back to M so this is a polygon required polygon guys so this is a required square J N okay let's try and measure the area so you had measured this area is a 54 this area is also 54 amazing isn't it so by this construction we could get a square which has same area as that of the given rectangle so hence we could convert a rectangle into a square of same area that's the construction now how did it all happen let's now look at the