Class 10 math (India)
Ratio in which a line divides a line segment
In an earlier video, we learned how to find the ratio in which a point divides a line segment. Here, let's learn how to find the ratio in which a line divides a line segment. Let's also learn how to find the point of intersection. Created by Aanand Srinivas.
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- how do we take out the coordinates if the points are in the ratio say... 2:3 i am getting a little confused(1 vote)
find the ratio in which the line segment joining 1 comma minus 5 and minus 4 comma 5 is divided by the x axis now we're not being given a point here can you see we've been given a line the x axis is a line so you define how this x axis divides this line segment but what does that really mean that means that you find where this x axis cuts this line segment that's a point and then see how that point divides in what ratio that point divides this line segment that's what the question is asking us in fact if we read further find the coordinates of that point of intersection where the x axis meets this line segment how do you want to do this let's first draw a diagram then let's look at the diagram to see what ideas we get and then let's go ahead and find the ratio and the coordinates let's start by creating the coordinate axis I would like you to see what happens when you draw this look at the diagram and see what you need to do without going into your mind like one of the ways to think about it is draw the diagram and then go okay how can I use section formula know what is M here what is n here and so on that is usually not the best way to think about it there are more intuitive ways to solve this so let's begin right now and do solve for the first point 1 comma minus 5 that's one to the right and then - 1 - 2 - 3 - 4 - 5 so I have 1 comma minus 5 over here 1 comma minus Phi and I need - 4 comma 5 that's 1 2 3 4 4 steps to the left and then 5 to the 5 above because it's + 5 1 2 3 4 5 somewhere over here - 4 comma 5 and our 1 the line segment that's connecting the two so that's this is the line segment that's gonna connect the 2 so that's the line segment how is it divided by the x axis and now we can see that the x axis does divide this line segment now it's totally possible that the x axis did not but can you see that you could have seen the question and been sure that the x axis divides it because one point has an why axis y coordinate value and the other one has a positive so if this line segment has to go somewhere below the x-axis to somewhere above it has to cut the x-axis right and I can oh I just noticed one more thing and you can notice it as well what is that this minus five is five steps below the x axis and plus five is five steps above the x axis so keep that in mind because I think that's gonna play a part in our answer over here because what is the question right now find the ratio in which the line segment joining these two points is divided by the x axis so where is the point at which the intersection is happening that's over here that looks like some minus one point five I think somewhere I don't know somewhere there a good diagram my teacher used to always say can actually help you guess the answer much before you actually have it so now you have this point and we don't know the coordinates of this point or do we we do know the y coordinate right it's the x axis so it's something comma zero so I'm gonna call it X comma zero it's something gamma zero we don't know it so now what we can do is see if we can find this X and then find the ratio or something but then I know something I know that finding the ratio is usually easier and why because to find the ratio I don't need all the coordinate values I don't need both x and y I only need X fully the x coordinate of all the three points or the y coordinate of all the three points if I have that I'm done and why is that we saw that a little bit in the previous video so if you want to take a deeper look at why that is you can go into the previous video it just involves having two similar triangles and noticing that to find the ratio of two sides you either need the heights or the lens one of them will do so in this case we don't have both we have only one we have the Y coordinates of all the points but that's enough so what do we do now we have to find this length by this length actually no we need to find this length by this length because the ratio the point starts the yellow point is given first so this one by this one and what is that going to be equal to that's going to be equal to this height this hype now why is this a quick refresher of why this is the case this is because this angle this triangle that we see over here has a 90 degrees over here and this triangle has a 90 degrees over here this angle is equal to is equal to this angle so let me draw that this angle and this angle are equal let's make this yellow yeah why are they equal is because there are alternate angles and because these two these two lines are parallel and this is a transversal they're alternate angles and now you've shown they're similar triangles you've not used any formula you just noticing the geometry of this and because it's some of the triangles you know that this side by this side will be the same as this side by this side but is what you want you want this side by this side now what is the length of this side the coordinate the y con is -5 so the length is 5 and what about this side the y coordinate is 5 which means the length is 5 again and that's interesting because not only are these two triangles similar now you know they are congruent because they have the same lengths now what does that mean that means that this side and this side are in the ratio 1 is to 1 or in other words this side and this side are also in the ratio 1 is to 1 which means the answer we want find the ratio the ratio is ratio is 1 is to 1 they are equal now notice that we actually found the ratio even without having to find the coordinates of the point of intersection yeah I mean one of the ways it at least when I think about it was like okay maybe we have to find this point first and only then I can find the ratio and that's not true you can find the ratio before you find the point because the ratio is easier but of course you can also find the point because once you had these two sites to be equal you know that these two sides are also equal and you can use that to find you know these two sides this one and this one are also equal and that's enough to find the coordinates of this point watch what happens when you pause the video and look at look at nothing else is needed you just know the coordinates of these and you know that these two lines are certain segments are equal that's all you need to find this x-value do it I'm gonna do it right now so I know that all the way from here to this end the length is going to be minus 4 2 plus 1 that's a length of 5 so that entire length is 5 but then I want only half of that because that's what at this point is this point is right in between those two right that's why did we call them congruent so this length is 2.5 so then I'll need to find my X's at 2.5 2 minus 4 and there I will have it so 2.5 2 minus 4 is -3 minus 2 and then minus 1 point 5 so minus 1 point 5 comma 0 I can just verify this by walking left words from plus 1 so 1 minus 2.5 should also give me the answer because this length is also 2.5 so 1 minus 2.5 is 0 minus 1 minus 1 point 5 and our diagram already suggested that because we drew it not not too badly I think so there you have it