Video: Finding the Coordinates of the Vertices of a Triangle after Rotation

Determine the coordinates of the vertices’ images of triangle 𝐴𝐵𝐶 after a counterclockwise rotation of 180°‎ around the origin.


Video Transcript

Determine the coordinates of the vertices’ images of triangle 𝐴𝐵𝐶 after a counterclockwise rotation of 180 degrees around the origin.

In this question, we’ll be performing a rotation. So we’ll be turning the shape. We’ll be rotating this triangle through an angle of 180 degrees. And we’re told to do this in a counterclockwise direction, although, for a 180-degree angle, it doesn’t matter whether the direction is clockwise or counterclockwise. The center of rotation here is the origin. That’s the coordinate zero, zero. So let’s go ahead and carry out this rotation.

Beginning with point 𝐴, we can see that this is located with an 𝑥-coordinate of negative eight and a 𝑦-coordinate of seven. So when we rotate this 180 degrees counterclockwise, the 𝑥-coordinate will be at eight and the 𝑦-coordinate will be at negative seven. In other words, there will still be a movement of eight units in the 𝑥-direction. It was eight units to the left or negative eight. And it’s now eight units to the right or positive eight. The 𝑦-coordinate will still be seven units away. It was positive seven units and it’s now negative seven units. We could label the image of vertex 𝐴 as 𝐴 prime.

Vertex 𝐵 on the triangle is at negative three, seven. So when we rotate it 180 degrees, it will still have an 𝑥-coordinate value of three, this time of positive three, and at a distance of seven in the 𝑦-axis, this time at negative seven. And we can label this new vertex as 𝐵 prime. As a check, we can notice that the line joining 𝐴 prime and 𝐵 prime is also horizontal in the same way as the line joining 𝐴𝐵. This is because our horizontal line which has been rotated 180 degrees would also produce another horizontal line. Our final vertex, 𝐶, is at negative four, three. So its image, 𝐶 prime, will be at the coordinate four, negative three. And we could complete the drawing of the triangle 𝐴 prime 𝐵 prime 𝐶 prime.

In a rotation, the object and its image will always stay the same size. So it’s worth checking some key lengths to see if they’re the same size as in the original object and its image. We can see that both the original shape and its image have a horizontal length of five units. And both triangles have a perpendicular height of four units. The question asked us to write the coordinates of the vertices’ images. So we can write 𝐴 prime as eight, negative seven, 𝐵 prime as three, negative seven, and 𝐶 prime is four, negative three. We can also see in this question that, in a rotation of 180 degrees about the origin, a point 𝐴 with coordinate 𝑥, 𝑦 will be rotated to give the image 𝐴 prime of coordinates negative 𝑥, negative 𝑦.

If we look at the original vertex 𝐴 with coordinate negative eight, seven, the image 𝐴 prime had the coordinate eight, negative seven. In the same way, the coordinate 𝐵 at negative three, seven became 𝐵 prime at three, negative seven. Coordinate 𝐶 at negative four, three became 𝐶 prime at four, negative three. After a rotation of 180 degrees about the origin, if the value of the 𝑥 was positive, it becomes negative and if it was negative, it becomes positive. The same is true for the 𝑦-value. This fact can be a helpful check whenever we’re carrying out this kind of rotation. Here, we can list our final coordinates for the answer.

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