Question Video: Rotating Shapes on a Coordinate Grid Mathematics • 8th Grade

Video Transcript

Rotate triangle 𝐴𝐵𝐶 180 degrees clockwise about the origin, and state the coordinates of the image.

In this question, we’re asked to perform a rotation of 180 degrees. When we’re doing a rotation of 180 degrees, it doesn’t matter whether the direction is clockwise or counterclockwise. The center of rotation is the origin. That’s the coordinate zero, zero. Sometimes, rotations can be difficult to visualize, so using tracing paper can often help. In this case, we would trace over triangle 𝐴𝐵𝐶, put the tip of our pencil in the center of rotation at the origin, and turn the tracing paper 180 degrees clockwise.

We can also perform a rotation by drawing lines from each vertex through the center of rotation. Beginning with vertex 𝐵, we can draw a ray through the center of rotation. The new vertex or the image of 𝐵 will lie on this line. Vertex 𝐵 is one unit across to the right and two units up. Therefore, the image will be one unit to the left, or negative one units, and two units down, or negative two units. So, the image of 𝐵, which we can refer to as 𝐵 prime, will be at the coordinate negative one, negative two.

Looking at vertex 𝐴, if we draw a line from 𝐴 to the center of rotation, this will be two units along. Therefore, the image of this point will also be two units from the center of rotation. Originally, 𝐴 was at negative two on the 𝑥-axis, and it will now be at positive two. So, the image of 𝐴, 𝐴 prime, will be at the coordinate two, zero.

The vertex 𝐶 is one unit across and three units down, or negative three units. So, the rotated vertex of 𝐶 prime will be one unit to the left, or negative one, and up three units. We can see that our rotated image, which we called 𝐴 prime 𝐵 prime 𝐶 prime, would appear as shown. When the center of rotation appears inside an object, then the image will overlap with it. We can list the coordinates of the image. 𝐴 prime would be at two, zero. 𝐵 prime is at negative one, negative two. And 𝐶 prime is at negative one, three.

To answer this question, we could also have used the rule that for a rotation of 180 degrees about the origin, the image of a coordinate 𝑥, 𝑦 would be negative 𝑥, negative 𝑦. In other words, if the coordinate had an 𝑥-value which was positive, the image would have a negative 𝑥-value. If it was originally negative, then the image would have a positive 𝑥-value. The same is true for the 𝑦-value.

We can see that the original coordinate 𝐴 was at negative two, zero. So, the image of this 𝐴 prime had a coordinate of two, zero. The coordinate of 𝐵 at one, two becomes 𝐵 prime at negative one, negative two. The coordinate of 𝐶 at one, negative three becomes 𝐶 prime at negative one, three. So, we can see that performing the rotation, either using tracing paper or lines through the center of rotation or by using the rule for rotation of 180 degrees about the origin, would give us the answer that the coordinates of the image are two, zero; negative one, negative two; and negative one, three.