Question Video: Understanding the Effect of a Rotation of a Triangle given the Angle of Rotation | Nagwa Question Video: Understanding the Effect of a Rotation of a Triangle given the Angle of Rotation | Nagwa

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

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What is the image of β³πΆπ·π after a β108Β° rotation?

02:47

### Video Transcript

What is the image of triangle πΆπ·π after a negative 108-degree rotation?

We are asked to rotate triangle πΆπ·π negative 108 degrees about point π. To do this, we note that the given polygon is a regular decagon. So, the angles around π are all congruent. We can also note that the sum of the angles around π 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 π 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 πΆπ·π, which are line segments. For example, we can rotate line segment ππΆ 36 degrees clockwise to coincide with line segment ππ΅, then another 36 degrees to coincide with line segment ππ΄, and a third rotation of 36 degrees to line segment ππ. Thus, the image of line segment ππΆ after a rotation of negative 108 degrees about π is line segment ππ.

We can do the same for another side of the triangle, line segment ππ·. Finding the images of these two sides should be sufficient information for us to identify the location of the image of triangle πΆπ·π. So if we rotate line segment ππ· negative 108 degrees about π, we get line segment ππ΄.

Therefore, when we rotate triangle πΆπ·π negative 108 degrees about π, the triangle will be rotated onto the third triangle clockwise from triangle πΆπ·π. We see that this is triangle ππ΄π. Thus, triangle ππ΄π is the image of triangle πΆπ·π.

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