Video: Determining the Rotation Used to Move a Vertex of a Triangle

A triangle graphed on the coordinate plane has a vertex at (6, 0). Which of the following rotations would move the vertex to the point (0, 6)? [A] 90°‎ clockwise around the origin. [B] 90°‎ counterclockwise around the origin. [C] 180°‎ clockwise or counterclockwise around the origin.

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Video Transcript

A triangle graphed on the coordinate plane has a vertex at six, zero. Which of the following rotations would move the vertex to the point zero, six? Option (A) 90 degrees clockwise around the origin. Option (B) 90 degrees counterclockwise around the origin. Or option (C) 180 degrees clockwise or counterclockwise around the origin.

We could start this question by sketching out where the original vertex would be. We’re told that this vertex is at six, zero. Let’s call this original vertex 𝐴. The image of this vertex after the rotation is at zero, six. Let’s call this 𝐴 prime. So how would we describe the rotation from 𝐴 to 𝐴 prime? We can recall that, to describe a rotation, we need to say three things, the angle, the direction, and the center of rotation. If we were to draw the angle between 𝐴 and 𝐴 prime, we would see that there’s a right angle of 90 degrees.

In terms of the direction, the direction that’s opposite to the one in which the hands turn on a clock is called counterclockwise, and that’s the direction that we can see here. The center of rotation will be the origin or the coordinate zero, zero. To put these three pieces of information together, we would say that this is a rotation of 90 degrees counterclockwise around the origin. But there is one other way in which we could have performed this rotation. And that is that we could have gone from vertex 𝐴 to 𝐴 prime by going in the opposite direction through an angle of 270 degrees.

Here, the center of rotation would have stayed the same, but we would have described the rotation as a 270-degree clockwise rotation around the origin. However, out of these two descriptions, only one appears in our answer options. And that is the one given in option (B), a rotation of 90 degrees counterclockwise around the origin.

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