Triangle ABC is similar to triangle HJK. If the coordinates of points A, B, and C are (2,5), (1,-2), and (-3,6), respectively, and the coordinates of points H and J are (3,2.5) and (0.5,-15), respectively, what are the coordinates of point K?
Here, we have a visual representation of triangle ABC and its points, and then the two points that we know from triangle HJK.
The following transcript is provided for your convenience.
We’re trying to find K. In order to find K, we first need to look at the way that these triangles relate to each other, how much bigger, smaller they are than each other since similar triangles have proportional, corresponding sides. So, I’m going to start by looking at AB. So, from point A to point B, the change in x is 1. From B to A, we just go to the right one, so the change in x is 1. The change in y – and that little triangle just means change in, the change in the y values for A to B is 7. The difference between 5 and -2 is 7.
Now, I’m going to look at the corresponding side to AB, which would be HJ, and see the x value and the y value of those points has changed.
So, now, from H to J, we’re going to look at the change in x, and the change in y. From H to J, the change in x, it goes from 0.5 to 3, so there’s a difference of 2.5, so the change in x is 2.5. The change in y from -15 all the way up to 2.5, so that’s 15, 17.5, the change in y is 17.5.
So, how do these values relate to each other? Well, 2.5 is 2.5 times 1, and 17.5 is also 2.5 times 7. So, the change in x and the change in y from triangle ABC to triangle HJK is 2.5 times bigger. So, to find K, we need to look at the corresponding side, BC, and JK could correspond to that, so I know K is going to be somewhere over here because it’s going to correspond to BC.
So, now we need to look at the change in BC, the change in x and the change in y. Since now we know that for triangle HJK, it’ll just be 2.5 times greater, so if we can find out what the change in x and the change in y for segment BC is, the same change times 2.5 will happen with JK.
So, from B to C, our x values are -3 and 1, so from -3 to 1, that’s a change in x of 4, and the change in y from -2 all the way up to 6, that’s a change of 8.
And again, the change for JK is going to be 2.5 times greater than that changes. So, the change for J to K, change in x is 2.5 times greater than the change from B to C. So, for B to C, the change in x was 4, 4 times 2.5 is 10, so the change in x there is going to be 10, and the change in y was 8 for B to C, so 2.5 times that, 8 times 2 is 16, half of 8 is 4, so that’s going to be 20.
So, that means that for J to K, we’re going to shift 10 to the left, and 20 up. So, 10 to the left would be -9.5, so there’s -9 here and a half, and then 20 up for -15, would positive 5. So (-9.5,5), which is about there, so this would be K, (-9.5,5).
That’s where K would be, and you could, of course, draw the triangle now that you know where K is.