Question Video: Understanding the Effect of a Rotation of a Triangle given the Angle of Rotation Mathematics • First Year of Preparatory School

Video Transcript

What is the image of triangle 𝐶𝐷𝑀 after a negative 108-degree rotation?

We are asked to rotate triangle 𝐶𝐷𝑀 negative 108 degrees about point 𝑀. To do this, we note that the given polygon is a regular decagon. So, the angles around 𝑀 are all congruent. We can also note that the sum of the angles around 𝑀 must be 360 degrees. Thus, each angle at the center has a measure calculated by 360 degrees divided by 10, which means each central angle has a measure of 36 degrees. Then, we note that negative 108 degrees equals three times negative 36 degrees. And we recall that rotating a negative value is always in the clockwise direction. So we can find angles of measure negative 108 degrees by rotating through three triangles in a clockwise direction.

We note that each rotation of negative 36 degrees about 𝑀 will rotate the triangles clockwise such that they are coincident. One way of seeing this is to consider the rotation of each side of triangle 𝐶𝐷𝑀, which are line segments. For example, we can rotate line segment 𝑀𝐶 36 degrees clockwise to coincide with line segment 𝑀𝐵, then another 36 degrees to coincide with line segment 𝑀𝐴, and a third rotation of 36 degrees to line segment 𝑀𝑃. Thus, the image of line segment 𝑀𝐶 after a rotation of negative 108 degrees about 𝑀 is line segment 𝑀𝑃.

We can do the same for another side of the triangle, line segment 𝑀𝐷. Finding the images of these two sides should be sufficient information for us to identify the location of the image of triangle 𝐶𝐷𝑀. So if we rotate line segment 𝑀𝐷 negative 108 degrees about 𝑀, we get line segment 𝑀𝐴.

Therefore, when we rotate triangle 𝐶𝐷𝑀 negative 108 degrees about 𝑀, the triangle will be rotated onto the third triangle clockwise from triangle 𝐶𝐷𝑀. We see that this is triangle 𝑃𝐴𝑀. Thus, triangle 𝑃𝐴𝑀 is the image of triangle 𝐶𝐷𝑀.