Question Video: Finding the Coordinates of the Vertices of a Triangle after Rotation Mathematics • First Year of Preparatory School

Video Transcript

Determine the images of the vertices of triangle 𝐴𝐵𝐶 after a clockwise rotation of 90 degrees about the origin.

So the key information within the question, we’re told that this rotation is about the origin. We’re also told that it’s a clockwise rotation by 90 degrees. We’ll look at two different methods for working out the solution to this problem. Our first method uses the coordinate grid in the diagram that we’ve been given. We’ll look at each of the three points in turn, beginning with point 𝐴.

I’ve connected point 𝐴 to the origin using horizontal and vertical lines, and I can see that to get from the origin to point 𝐴, I need to go eight units to the left and then four units down. Let’s think about what happens when these two lines are rotated through 90 degrees clockwise. Well, the horizontal line of eight units is now a vertical line of eight units, and the vertical line of four units is now a horizontal line of four units. This tells us the image of the point 𝐴, 𝐴 prime. So we have the coordinates of 𝐴 prime: negative four, eight.

Now let’s repeat the same process for points 𝐵 and 𝐶. I draw in the horizontal and vertical lines connecting the origin to point 𝐵, both of which are of length three units. I then rotate this line through an angle of 90 degrees clockwise, and now I can see the image of point 𝐵. So we have the coordinates, 𝐵 prime is negative three, three.

Finally, let’s do the same thing for point 𝐶. I draw in the horizontal lines connecting the origin to point 𝐶, that is three units across and seven units down. I rotate this pair of lines again through 90 degrees in a clockwise direction about the origin, and now I have the image of point 𝐶. 𝐶 prime is negative seven, three.

By joining the three points together, I can now see the image of the original triangle 𝐴𝐵𝐶 on the coordinate grid. So that was our first method, using the diagram itself. Now if I write down the coordinates of the original points 𝐴, 𝐵, and 𝐶, you may be able to see a shortcut for the second method.

So compare the coordinates of 𝐴, 𝐵, and 𝐶 with the coordinates of the image 𝐴 prime, 𝐵 prime, and 𝐶 prime. The 𝑦-coordinates of the original point are in fact the 𝑥-coordinates of the images. This isn’t a coincidence. And so as a general rule, we can say that the point with coordinates 𝑥, 𝑦 will be mapped to a point whose 𝑥-coordinate is now 𝑦.

Now is it also true that the 𝑦-coordinate of the image is the same as the 𝑥-coordinate of the original point? Well, not quite but nearly. Looking at the 𝑦-coordinates of the image and the 𝑥-coordinates of the original point, we can see that they’ve actually been multiplied by negative one. Negative eight has become eight, and negative three has become three. Therefore, if we’re mapping the general point 𝑥, 𝑦 using this rotation, we can see that the 𝑦-coordinate of the new point will be negative 𝑥.

This is a general rule for performing a clockwise rotation of 90 degrees about the origin. The point with coordinates 𝑥, 𝑦 will get mapped to the point with coordinates 𝑦, negative 𝑥. So if you can remember this general rule, then this would be an alternative method of performing this type of rotation. Both methods will of course arrive at the same answer.

The coordinates of 𝐴 prime are negative four, eight; the coordinates of 𝐵 prime are negative three, three; and the coordinates of 𝐶 prime negative seven, three.