Video Transcript
Which of the following coordinates represent the line segment 𝑋𝑌 after a geometric translation of 𝐴𝐵 in the direction of ray 𝐴𝐵, where 𝐴 has coordinates negative three, negative two and 𝐵 has coordinates negative one, negative six, given that 𝑋 has coordinates one, four and 𝑌 has coordinates three, negative one? Option (A) 𝑋 prime with coordinates three, zero, 𝑌 prime with coordinates five, negative five. Option (B) 𝑋 prime with coordinates negative three, six, 𝑌 prime with coordinates five, negative five. Option (C) 𝑋 prime with coordinates three, zero, 𝑌 prime with coordinates negative one, one. Or option (D) 𝑋 prime with coordinates negative one, eight, 𝑌 prime with coordinates one, three.
In the figure, we can see that the line segment 𝑋𝑌 has been drawn on the coordinate grid. And we are told that this line segment is translated. We can recall that any translation can be thought of as a change in the horizontal and vertical displacement. And generally this will be given to us in these terms.
However, here, we are told the translation in terms of a ray from 𝐴 to 𝐵. Let’s consider this ray 𝑃𝑄. As we know the direction — we’re going from 𝑃 to 𝑄 — then we can see how knowing two points and the direction of travel between them will allow us to determine the horizontal and vertical displacement that this must be. Whatever the displacement is on the given ray 𝐴𝐵 in the question will be the same displacement, or the same translation, that we must apply to the line segment 𝑋𝑌. And it might be useful to draw out this line segment 𝑋𝑌 onto a larger grid so we can fit in all the important coordinates onto it.
So here we have a copy of the grid with the line segment 𝑋𝑌. We can then plot the points 𝐴 at coordinates negative three, negative two and 𝐵 at coordinates negative one, negative six. We can join 𝐴 and 𝐵 to create the ray 𝐴𝐵.
We can now determine the horizontal and vertical displacement between 𝐴 and 𝐵. Since we are going from 𝐴 to 𝐵, the horizontal displacement will be two units. And this is a positive value of two units because the direction is to the right. We can also see this if we use the coordinates of 𝐴 and 𝐵 directly. Because we are going from point 𝐴 to point 𝐵, we subtract the 𝑥-coordinate of 𝐴 from that of 𝐵, which also gives us two. This is the horizontal displacement of the ray 𝐴𝐵.
Now let’s do the same to find the vertical displacement. Using the graph, we can see that this is a vertical displacement of four units downwards. Using the coordinates, we could calculate the 𝑦-coordinate of 𝐵 subtract the 𝑦-coordinate of 𝐴, which gives us a vertical displacement of negative four. And the negative here would indicate that the direction is downwards, as we see on the graph. If the vertical displacement from these coordinates was positive, the translation would be upwards.
Now, we can apply the translation of two units right and four units downwards to the line segment 𝑋𝑌. And we can do this by taking each point in turn. Taking the point 𝑋, it moves two units to the right and four units down. As this point is the image of 𝑋, we can label it 𝑋 prime, and it has the coordinates three, zero. We can then perform the same translation of two units right and four units down on point 𝑌, which gives us the new point 𝑌 prime at the coordinates five, negative five.
Joining these coordinates would create the new line segment 𝑋 prime 𝑌 prime, which is the same length and at the same orientation as the original line segment 𝑋𝑌. And so the point 𝑋 prime at the coordinates three, zero and point 𝑌 prime at coordinates five, negative five is the answer to the question and also the answer given in option (A).
