Video: Finding the Coordinates of a Point given the Distance between It and Another known Point

Consider 𝐴(βˆ’1, βˆ’2) and 𝐡(βˆ’7, 7). Find the coordinates of 𝐢, given that 𝐢 is on the ray 𝐴𝐡 but NOT on the segment line 𝐴𝐡 and 𝐴𝐢 = 2𝐢𝐡.

02:46

Video Transcript

Consider 𝐴-coordinate negative one, negative two and 𝐡-coordinate negative seven, seven. Find the coordinates of 𝐢, given that 𝐢 is on the ray 𝐴𝐡, but not on the segment 𝐴𝐡 and 𝐴𝐢 equals two 𝐢𝐡.

So let’s start this question by considering our two coordinates. If it’s helpful, we could plot these two points on a coordinate grid. But sometimes, it’s sufficient just to plot their relative position to each other. Here, we’re told that there’s a ray 𝐴𝐡. That means that the line starts at 𝐴, goes through 𝐡, and continues indefinitely. We’re told that there’s a coordinate 𝐢, which is on the ray 𝐴𝐡, but not on the segment 𝐴𝐡, which means that 𝐢 isn’t between 𝐴 and 𝐡. So we can draw it on the line beyond 𝐴𝐡. We’re told that 𝐴𝐢 equals two 𝐢𝐡. This means that if we said π‘₯ was the length of 𝐢𝐡, then 𝐴𝐢 must be two times that, giving us two π‘₯. At this point, we don’t know the length π‘₯ or the length 𝐢𝐡. But given we know the coordinates of 𝐴 and 𝐡, we could work out the length 𝐴𝐡, which would also be of a length π‘₯.

An alternative way to consider this would be to think that the ratio of 𝐴𝐡 to 𝐡𝐢 would be one to one. So let’s now consider our length 𝐴𝐡. And we can do this by considering how we go from 𝐴 to 𝐡 with our coordinates. If we look at the π‘₯-coordinate of point 𝐴 as negative one and we go to our π‘₯-coordinate of 𝐡, which is negative seven. This means that we would move negative six horizontally. If we consider our 𝑦-values, then we have negative two on our 𝐴-coordinate up to seven on our 𝐡-coordinate, which represents a move of nine vertically. So now as we know that the ratio of 𝐴𝐡 to 𝐡𝐢 is one to one. This means that the journey from 𝐡 to 𝐢 must also be negative six horizontally and nine vertically. So to find our π‘₯-value in coordinate 𝐢, we go from the π‘₯-value in 𝐡, which is negative seven. And we subtract six since we had a negative six movement horizontally. So our π‘₯-value would be negative 13. For our 𝑦-values then, our 𝑦-value of 𝐡 was at seven. And we must add nine since we move nine vertically. And since seven add nine is 16, this gives us the 𝑦-value of 16.

So the coordinate of 𝐢 is negative 13, 16.

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