Question Video: Finding the Sum of the Coordinates of Two Points Where One Is Original and the Other Is an Image Point by Rotation Mathematics • 8th Grade

Video Transcript

If the image of point 𝑥, 𝑦 by rotation about the origin point with an angle measuring 270 degrees is 𝑎, 𝑏, find the value of 𝑎 plus 𝑦 and 𝑏 plus 𝑥.

Now, this question may seem a little confusing at first. But let’s consider that we are told that we have this general coordinate 𝑥, 𝑦. For the purposes of understanding this question, we could plot 𝑥, 𝑦 anywhere. But let’s plot it here in the first quadrant. We are then given that this point 𝑥, 𝑦 is rotated about the origin point. So the center of rotation is at the coordinates zero, zero.

The angle of rotation is 270 degrees. And when no direction is specified, we use the convention that the direction must be counterclockwise. So the point 𝑥, 𝑦 is rotated to this point. We can see that this turn is counterclockwise through 270 degrees, or three 90-degree turns, to here.

We are given that this new point, the image of the coordinates 𝑥, 𝑦, has the coordinates 𝑎, 𝑏. But is there another way in which we can relate this new coordinate to the original coordinate and write it using 𝑥 and 𝑦? And yes, there is, by recalling this property. A rotation of 270 degrees counterclockwise about the origin is equivalent to the coordinate transformation such that the coordinates 𝑥, 𝑦 map to the coordinates 𝑦, negative 𝑥. So the coordinates a, 𝑏 are equivalent to the coordinates 𝑦, negative 𝑥. That means we can equate the 𝑥-coordinates to write that 𝑎 is equal to 𝑦. And equating the 𝑦-coordinates, we have that 𝑏 is equal to negative 𝑥.

We now need to find the values of 𝑎 plus 𝑦 and 𝑏 plus 𝑥. Let’s take 𝑎 plus 𝑦 first. We’ve already worked out that 𝑎 is equal to 𝑦. So 𝑎 plus 𝑦 is equivalent to 𝑦 plus 𝑦, which is two 𝑦. And that’s the answer to this first part done. Of course, we could also write that 𝑎 plus 𝑦 is equivalent to two 𝑎. But let’s keep our answers in terms of 𝑥 or 𝑦.

Now let’s do the same to find the value of 𝑏 plus 𝑥. This time, we know that 𝑏 is equal to negative 𝑥. And negative 𝑥 plus 𝑥 would give us zero. Therefore, we can give the answers to both parts of the question, as 𝑎 plus 𝑦 equals two 𝑦 and 𝑏 plus 𝑥 equals zero.