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Dividing line segments: graphical

Watch Sal figure out the coordinates of a point between two other points that give a certain ratio. A graph is given to make it easier to visualize the problem. Created by Sal Khan.

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  • female robot grace style avatar for user Amanda Aliberto
    at , i don't understand how you got 1/4 for the ratio 3:1?

    i've been doing the lesson this video is for in geometry for awhile now, and the others don't make sense to me either. i started to think it was just getting them to 4:4, since 1 + 3 = 4 and since there was already a 4 in 1/4, you just made it a 1 in 3:1.

    but that doesn't always work. for example, problems with the starting ratio of 3:4 for AB had another one of 3:7 or 3/4. i just don't get it.
    (51 votes)
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    • aqualine ultimate style avatar for user Grace Miedziak
      I'll try to clarify, I am studying Geometry myself but I would like to help :)

      First, let's start with an example using ratios. Say that a cookie recipe said that the ratio between flour and sugar is 3:1 cups. That means for every 3 cups of flour, there is 1 cup of sugar. After you add 3 cups of flour and 1 cup of sugar in, you just added in 4 total cups of ingredients! So, if I need to add 3 cups of flour and 1 cup of sugar, and I start by adding 1 single cup of flour, I am now 1/4 finished with putting the ingredients in (because I still need to add 2 more cups of flour and 1 sugar).

      Although using this example for this situation may be a bit more confusing, it still works. Like Sal said at , the longer part of the line segment is three times as large (or 3x, as he states it) than the smaller part, x. It's like how we add 3 times as much flour than sugar in the cookie recipe. Now that we know CB is x, and BA is 3x, we can say x + 3x, or x+x+x+x, equals 4x, or 4 units. So, CB is 1/4.

      I hope this helped! XD
      (63 votes)
  • male robot hal style avatar for user Hassan Saleh
    How can we get the soution mathematically?
    (21 votes)
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  • starky tree style avatar for user jk parker
    is there a way to understand what fraction to use? Why dont we just use the ratio? How do we use the ratio to find the fraction to use in our equation? what if the numbers are different,say 3:4 instead of 3:1, is the fraction then 1/7? I dont understand.
    (9 votes)
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    • ohnoes default style avatar for user TheMightyJakjak
      This one is a little tricky on the first go. The reason they use "1/4" is because a 3:1 ratio is 3 to 1 distance on the line segment given. On a 3:4 ratio, the fraction would would be "3/7", because it would be 3 parts out of 7 total parts on the line segment.

      Hope this could clarify!
      (14 votes)
  • piceratops ultimate style avatar for user Gaycel Sotelo
    can i ask what formula will usefor finding ratio?
    thanks
    (15 votes)
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  • aqualine ultimate style avatar for user Azaryah
    How do I tell if it wants me to do C vertical to A horizontal and vice versa because its so confusing
    (6 votes)
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    • marcimus pink style avatar for user Laurie Metzger
      So each point has an x coordinate and a y coordinate. You want to first look at one and then the other. So if you first look at the x coordinates, you can see that the total length is 16 (the two points are sixteen dashes apart on the x axis). Then you look at the y coordinates for both points, and see the total height is 5 (the points are five dashes apart when looking at the y axis.

      We know the ratio is 3:1, so since 3 + 1 equals 4, we want to divide these numbers into four parts. 16 divided into four parts is 4. And 4 divided into four parts is 1. We want the line AB to have 3 of the parts and the line BC to have one of the parts. That will make the line AB to be three times as long as BC.

      So if we start at point C, we want to count up 1 and over 4 to get point B. Or we could start at point A and count down 3 and to the left 12. Either way, you get point B in the same place and create a line AB which is three times as long as BC (so point B is 1/4 the distance of the line of AC).
      (7 votes)
  • blobby green style avatar for user kr032130
    how does this make sense?
    (8 votes)
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  • orange juice squid orange style avatar for user shamkirb
    Where did you get 1/4 from?
    (5 votes)
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  • old spice man blue style avatar for user montahar
    How is it that point B is 1/4th of the way across point A and C? Is it because the whole thing is 4?
    (3 votes)
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    • female robot grace style avatar for user StudyBuddy
      It's 1/4 of the way from point C to point A. This is because the ratio of AB to BC is 3 to 1. And if you want to convert a ratio to a fraction, you need to take the 1 for the numerator and then add the two together for the denominator. I hope this helped! I'm gonna include Grace's answer to a similar question. Mabye it'll explain it a little more.
      Grace's answer:
      I'll try to clarify, I am studying Geometry myself but I would like to help :)

      First, let's start with an example using ratios. Say that a cookie recipe said that the ratio between flour and sugar is 3:1 cups. That means for every 3 cups of flour, there is 1 cup of sugar. After you add 3 cups of flour and 1 cup of sugar in, you just added in 4 total cups of ingredients! So, if I need to add 3 cups of flour and 1 cup of sugar, and I start by adding 1 single cup of flour, I am now 1/4 finished with putting the ingredients in (because I still need to add 2 more cups of flour and 1 sugar).

      Although using this example for this situation may be a bit more confusing, it still works. Like Sal said at
      , the longer part of the line segment is three times as large (or 3x, as he states it) than the smaller part, x. It's like how we add 3 times as much flour than sugar in the cookie recipe. Now that we know CB is x, and BA is 3x, we can say x + 3x, or x+x+x+x, equals 4x, or 4 units. So, CB is 1/4.

      I hope this helped! XD
      Me again:
      I hope this helps! Please do let me know if it doesn't! :)
      (7 votes)
  • blobby green style avatar for user federico-nicolas
    I don't understand the ratios part, I've got an exercise where they say 5:6 is 🟥🟥🟥🟥🟥🟦 5/6 or 1/6, but the guy on this video says 3:1 is 🟥🟥🟥🟦 3/4 or 1/4
    (3 votes)
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  • hopper cool style avatar for user ALYSIQUE UNIQUE = )
    I do not understand on how to know whether the coordinate is on the upper half of the line segment or the lower half of the triangle. It seems to always switch around on the questions, with no apparent pattern!
    (3 votes)
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    • hopper cool style avatar for user James Lu
      Hi!
      Good question, you can do this by comparing the point on the triangle to the midpoint (of the hypotyneus.) The midpoint formula (there is a video explaining it above), is (x1+x2 /2 , y1+y2/2), where the x's and the y's are the coordinates of the points. After finding the midpoint, you can compare it to the desired point you have on the triangle based off whether the midpoint's y axis is greater or less than the selected point.
      Hope this helps!
      (3 votes)

Video transcript

Find the point B on segment AC, such that the ratio of AB to BC is 3 to 1. And I encourage you to pause this video and try this on your own. So let's think about what they're asking. So if that's point C-- I'm just going to redraw this line segment just to conceptualize what they're asking for. And that's point A. They're asking us to find some point B that the distance between C and B, so that's this distance right over here. So if this distance is x, then the distance between B and A is going to be 3 times that. So this will be 3x. That the ratio of AB to BC is 3 to 1. So that would be the ratio-- let me write this down. It would be AB-- that looks like an HB-- it would be AB to BC is going to be equal to 3x to x, which is the same thing as 3 to 1, if we wanted to write it a slightly different way. So how can we think about it? You might be tempted to say, oh, well, you could use the distance formula to find the distance, which by itself isn't completely uncomplicated. And then this will be 1/4 of the way. Because if you think about it, this entire distance is going to be 4x. Let me draw that a little bit neater. This entire distance, if you have an x plus a 3x, is going to be 4x. So you'd say, well, this is 1 out of the 4 x's along the way. This is going to be 1/4 of the distance between the two points. Let me write that down. This is 1/4 of the way between C and B, going from C to A. B is going to be 1/4 of the way. So maybe you try to find the distance. And you say, well, what are all the points that are 1/4 of the way? But it has to be 1/4 of that distance away. But then it has to be on that line. But that makes it complicated, because this line is at an incline. It's not just horizontal. It's not just vertical. What we can do, however, is break this problem down into the vertical change between A and C, and the horizontal change between A and C. So for example, the horizontal change between A and C, A is at 9 right over here, and C is at negative 7. So this distance right over here is 9 minus negative 7, which is equal to 9 plus 7, which is equal to 16. And you see that here. 9 plus 7, this total distance is 16. That's the horizontal distance change going from A to C, or going from C to A. And the vertical change, and you could even just count that, that's going to be 4. C is at 1. A is at 5. Going from 1 to 5, you've changed vertically 4. So what we can say, going from C to B in each direction, in the vertical direction and the horizontal direction, we're going to go 1/4 of the way. So if we go 1/4 in the vertical direction, we're going to end up at y is equal to 2. So I'm just going, starting at C, 1/4 of the way. 1/4 of 4 is 1. So I've just moved up 1. So our y is going to be equal to 2. And if we go 1/4 in the horizontal direction, 1/4 of 16 is 4. So we go 1, 2, 3, 4. So we end up right over here. Our x is negative 3. So we end up at that point right over there. We end up at this point. This is the point negative 3 comma 2. And if you were really careful with your drawing, you could have actually just drawn-- well, actually you don't have to be that careful, since this is graph paper. You actually could have just said, hey, we're going to go 1/4 this way. Where does that intersect the line? Hey, it intersects the line right over there. Or you could have said, we're going to go 1/4 this way. Where does that intersect the line? And that would have let you figure it out either way. So this point right over here is B. It is 1/4 of the way between C and A. Or another way of thinking about the distance between C and B, which we haven't even figured out. We could do that using the distance formula or the Pythagorean theorem, which it really is. This distance, the distance CB, is 1/3 the distance BA. The ratio of AB to BC is 3 to 1.