Video: Finding the Coordinates of a Point That Divides a Given Line Segment

Consider points 𝐴(2, 3) and 𝐡(βˆ’4, βˆ’3). Find the coordinates of 𝐢, given that 𝐢 is on the ray 𝐡𝐴 but not on the segment 𝐴𝐡 and 𝐴𝐢 = 2𝐴𝐡.

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Video Transcript

Consider points 𝐴 two, three and 𝐡 negative four, negative three. Find the coordinates of 𝐢 given that 𝐢 is on the ray 𝐡𝐴 but not on the segment 𝐴𝐡 and 𝐴𝐢 equals two 𝐴𝐡.

So the first thing I’ve drawn is I’ve drawn the points 𝐴 and 𝐡, and then also what we’ve drawn here is the line segment 𝐴𝐡. But now what we need to do is think about where the point 𝐢 is. Well, we’re told that it’s on the ray 𝐡𝐴 but not on the line segment 𝐴𝐡. But what does the ray 𝐡𝐴 mean? Well, if we know that 𝐢 is on the ray 𝐡𝐴, this means that from 𝐡 going towards 𝐴, 𝐢 is gonna lie along this line somewhere. However, we know that it’s not on the segment 𝐴𝐡, so we know that it’s not in our line segment 𝐴𝐡. So that means it’s not where the blue area is, so we know that it must extend beyond 𝐴.

So what I’ve drawn here is a sketch of the scenario, just looking at the idea of a line. So we’re told that 𝐴 to 𝐢 is equal to two 𝐴𝐡. So therefore, we know that our line is gonna be in the ratio one to two. And that’s if we have the points 𝐡, 𝐴, and 𝐢 in that order. Now, the way we can approach this is to look at our π‘₯-coordinates and then look at our 𝑦-coordinates. So we start by looking at our π‘₯-coordinates. So what we’ve done is labeled our coordinates π‘₯ sub two, 𝑦 sub two and π‘₯ sub one, 𝑦 sub one. I’ve just done it in this way around because we’re going from 𝐡 to 𝐴.

So to find the difference between our π‘₯-coordinates, what we’re gonna do is π‘₯ sub two minus π‘₯ sub one, which is gonna be equal two minus negative four. Well, if we subtract a negative, it’s the same as adding a positive, so we get two plus four which is equal to six. So we know that the difference between 𝐡 and 𝐴 in the π‘₯-coordinate is six units. So now if we consider our ratio, which is one to two, we can see that the difference between the π‘₯-coordinates between 𝐴 and 𝐢 is gonna be twice that of 𝐡 to 𝐴, so we make two multiplied by six. So therefore, it’s going to be 12.

So therefore, if we want to find out π‘₯ which we’re calling the π‘₯-coordinate of 𝐢, then this is going to be the π‘₯-coordinate of 𝐴 which is two plus 12 because that’s the difference between them, which is going to be equal to 14. Okay, great, so now we’re gonna look at the 𝑦-coordinates. So now what we want to do is find the difference between the 𝑦-coordinates. It’s gonna be 𝑦 sub two minus 𝑦 sub one, which is gonna be three minus negative three. Well, if you subtract a negative, it’s the same as adding a positive like we said before.

So once again, we can see the difference this time between the 𝑦-coordinates is six. So therefore, using the same rationale as before, that’s our ratio one to two, we can see that, therefore, the difference between the 𝑦-coordinates of 𝐴 and 𝐢 must be double that. So it’s gonna be 12. So therefore, the 𝑦-coordinate of 𝐢 is going to be three, which was the 𝑦-coordinate of 𝐴 plus 12 which is gonna be equal to 15. So therefore, we can say that the coordinates of point 𝐢 given that 𝐢 is on the ray 𝐡𝐴 but not on the segment 𝐴 to 𝐡 and 𝐴𝐢 is equal to two 𝐴𝐡 are 14, 15.

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