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Ratio in which a point divides a line segment

Let's learn how to find the ratio in which a point divides a line segment. Let's learn how to do this intuitively without having to rely on a formula to do it. Created by Aanand Srinivas.

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Video transcript

find the ratio in which the line segment joining the points minus 3 comma 10 and 6 comma minus 8 is divided by minus 1 comma 6 now there is some line segment that joins these two points this point divides that line segment we have to find what is the ratio in which it divides it now I'm taking the word of the question setter that this point is on this line segment because it could not it's possible that this point is not even on that line segment but I'm assuming that the question setter has already taken care of that assuming that's the case I need to find the ratio in which it divides it now even before I start thinking of this question like even to start doing that I need a picture of my coordinate axis because all questions are in this unit need the coordinate access to even start thinking about them and the first thing I want to do is notice or draw for myself where minus 3 comma 10 is and 6 comma minus 8 is so I'm going to do that minus 1 minus 2 minus 3 1 2 3 4 5 6 7 8 9 10 this is where approximately my minus 3 comma Denis let's just mark that minus 3 comma 10 and where is 6 comma minus 8 gonna be 1 2 3 4 5 6 - 1 - 2 - 3 - 4 - 5 6 7 8 that's approximately where that's gonna be so 6 comma minus 8 or negative 8 usually so now what do I want what is the line segment joining these two I can just draw that line segment over here that's a straight line so let's draw it this way that's the line segment joining the two of them not bad now what do I need to do - 1 comma 6 where is that going to be - 1 2 3 4 5 6 not bad this actually shows that our diagram is not too super unreasonable because I can see now that - 1 comma 6 looks as if it will be on this line and just by arriving here I can see that I have a good idea of what my answer should be right it's some like this length by this lid that's what they asking me right find the ratio in which the line segment joining these two points it is divided by minus one comma six is a long way of asking can you find this length by this length now how do we do something like this I want this length and I want that to be divided by this length over here now the moment I saw this question and I tried to solve it my instinct was to say okay you want this distance by this distance I know distance formula I can just use distance formula because I know this point and this point I can use distance formula to find this length x2 minus x1 squared plus y2 minus y1 squared the route whole root of that will give me that and I can do the same thing over here to find this length then I just have to divide the two and if you if you don't remember the distance formula you can always just take this length this length and then use Pythagoras theorem that is the derivation of distance formula the derivation of distance formula so you can do that but and I encourage you to do it actually that's that's how I did it at least in the beginning and that's not a wrong way to do it it's just a long way to do it because you're being asked just the ratio of these two line segments the lengths of these two line segments you're not being asked to find the lens so why do you want to do more work the necessity we do want to find just what's being are straight like we want the Lazy solution so is there a clever way to find the ratio of this line segment by this line segment without actually finding the lens themselves so when you're asked you'll say hey I know the ratio I don't know the lens but I know the ratio how do you do that now the clue is that this is a coordinate geometry problem which means that all our distances are basically measured either horizontally or vertically right that's what we mean that's why these are called rectangular coordinates so in when you have coordinates like this where you're given like minus 3 comma 10 what is this really say this is just saying that this point we have over here it's address is minus 3 comma 10 or in other words it's 3 units away distance from this line towards the left that's why there's our negative sign and it's 10 units away from this line vertically right so can I do something to make it so that I have vertical and horizontal lines in my way of approaching this question because my whole problem here the reason I'm not solving this quickly is that I have slanting lines I need to find the ratio of this line by this line but if I can somehow get it in terms of vertical lines or horizontal lines my my job will become much easier because those distances are already given to me so in an attempt to do that what I will do is try and extend this over here extend this line down way downwards and this towards the left now why I'm doing it is trying to see if I can connect in some way this length to some vertical or horizontal lengths and I'm gonna do the same with this one and also connect this now I've made some progress if you can see right because what I wanted was this line segments length by this line segments length but now that's the same as asking can you find the ratio of this side of my triangle here by this side of the triangle and I like triangles I know a little bit about triangles so maybe I have a better chance let's let's take away the coordinate axis for a bit so we can focus on what we have over here I have two triangles and I need to find the ratio of their sites and I know that whenever I think of ratio of sides of two triangles I begin to hope that they are similar because if they are then the ratio of this length by this length will be the same as the ratio of this length by this length or this length by this length right that's the definition of similar triangles so if I can show that these two triangles are similar I make like huge progress and this looks in this looks encouraging because these two triangles do look as if they would be similar and why am i why am I feeling so optimistic is because I already know this angle is 90 degrees and I know that this angle is 90 degrees this angle is 90 degrees and I can I now need to just look for one more angle to be equal and then I have a a similarity and I can think I'm gonna pick this one let's see I have this angle here and I have this angle over here and why are these those two look really equal but that's not enough can I explain why they are equal I can notice that this line and this line are definitely parallel because both are horizontal lines they're both parallel to this x-axis and this actually is just a transversal that's cutting these two parallel lines which means this angle and this angle are corresponding angles so this triangle is similar to this triangle which means our job our life has become much easier I don't need to care any more about this line segment by this line segment ratio I just care about this length by this length and that's much easier to find so let's stop right now and I would like you to find this length find this length and that's that divide the - that's your ratio so let's do it so what is this length gonna be finding vertical lengths in coordinate geometry just boils down to subtracting the y-coordinates because these just give you the vertical lengths anyway right so having this this 10 just means this entire length is 10 and 6 over here just means this distance is 6 so the gap between the two will be 10 minus 6 or 4 so that's the length of this side what about the length of this side similar story you have this 8 units below the x-axis and this is 6 units above the x-axis so if I have to like measure this whole distance this will be minus 8 this will be 6 and if I add the 2 I will get 14 as my length of this entire line 14 which means I already have my ratio it is 14 by 4 sorry forward by 14 the the order does matter right the order matters because is this by this is definitely not equal to this reciprocal so we have to depending on how they're asking the question which points coming first I need to answer it that way as well like if minus 3 comma 10 is given first I have to put this ratio first so now I have four by fourteen or two by seven let's use yellow equal to two over seven I can also use the language of ratios to write this you can maybe write this as 2 is to seven now you may have wondered why did we pick the y-axis I mean you could have picked the x-axis I mean it's it's either of the two which so if you have a preference for X specifically then you can pick X I just thought the Y looked bigger over here so I just picked it but you can verify you know if you're an exam sitting and you want to be doubly sure then you can verify whether your answer is correct by also checking with the x-axis because the answer it can't be different so what is this length this length is the x coordinate difference it's minus 1 over here minus 3 so the distance to walk from minus 1 to minus 3 is just 2 so this length is 2 what about this length it's 6 over here 2 minus 1 so to walk all the way to 0 from 6 and then walk another one to go 2 minus 1 so you have 7 over your 7 so you are getting the same ratio if you notice directly in fact in this case 2 by 7 you don't even have to divide by 2 so 2 over 7 or 2 is to 7 now what we did here is a somewhat unusual way to do this type of a problem because the most common approach I've seen as to take the ratio that we want as M is to n and use the section formula for either the x coordinates or the Y because in this case we have both given to us and get the answer now that's fine you can totally do that if you want to I just find this to be much more intuitive and easy to follow and I generally don't like to just be plugging things into a formula because it doesn't let me see what's going on or you can also sometimes a cleverly instead of taking the ratio as M is 2 n you can take it as K is 2 1 and solve the problem now I want to tell you that actually what we did here is the section formula but without using it it is the derivational section formula for all practical purposes if you'd kept these as variables you are actually just deriving the section formula and if this seemed a little bit long to you let me show you that the first couple of times you're noticing oh I need some of the triangles and so on but after that you'll do this really quickly how because you will read this question and you go okay the two points are minus 3 comma 10 and 6 comma minus 8 and they divided by minus 1 comma 6 you don't need like because you you don't need both the coordinates you can just pick one of them you know that so you'll just pick say the X and say how far is this point from this point x coordinate wise minus 3/2 minus 1 the distance is 2 so let's say 2 how far is the x coordinate of this point from this point you'll say ok 6 2 minus 1 that's 7 units away to go from 6 to minus 1 I have to walk 7 to 7 units I'm done that's it we could've even deleted some information from this if I had not given the Y coordinates at all you can see you can still solve the problem I just have to say it's minus 3 comma something 6 come or something and minus 1 comma something or I could have flipped it just given the Y coordinates and that's enough this question actually has way more information than needed to solve the problem