How to Calculate Rotation in Geometry

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Calculating rotation in geometry is something that you will have to do using a very specific equation. Calculate rotation in geometry with help from an experienced math professional in this free video clip.

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Video Transcript

Hi. I'm Drew Moyer and this is how to calculate rotation in geometry. Imagine that I tie up an object with a piece of rope and then begin to swing that object all the way around my body. This is analogous to what a rotation in geometry really is. And the object that I've tied up is the shape that's rotating. The rope is the radius of a circle and my body is the center of that circle. So let's take a look at our diagram. We have triangle 1. This is the original triangle and it's rotated to this position. And our job is to find out the measure of this angle and I'll call it theta. The first step is to find out the center of this circle which is the point about which this triangle is rotating. And the way I can do that is by extending both bases of both triangles and finding the point that they intersect. But first I'm going to have to do some work. I'm going to have to find out the slopes of those lines and I'm going to have to use that to find the equations of those lines. And then I'm going to have to solve the system of equations to find out where those lines actually intersect. So I've done some of the work for you. I found that the slope of the base of triangle 1 is zero, which is nice because that means I know that it's a horizontal line. I'll be able to use that later. Then for triangle 2 it was a little bit more complicated but I found that the slope is the square root of 3. So then I take that over to my equations work and I'm going to be using point slope form to find out the equations of both of these lines. So again, since the base of triangle 1 is a horizontal line I know that the equation for that must be y equals 1 and for the base of triangle 2, again a little more complicated but once I simplify all the work I get that y equals root 3x minus root 3 plus 1. And from there all I'm going to do is set them equal to each other, or excuse me, I'm going to put them under each other. And I'm going to solve by elimination. So again, I'm going to put triangle 2's line right above triangle 1's line. And I'm literally going to subtract out everything. So I do y minus y which is zero, and then I'm going to subtract this 1 from this 1 which means that they are going to cancel out along with the ys and I'm left with zero equals root 3x minus root 3. And from there I would just bring the root 3 back over to the other side and I would divide by root 3 to find that x equals 1. So now I know the x coordinate of this intersection and the y coordinate again, because the base of triangle 1 is a horizontal line. The y value is always going to be 1. So I know that this point right here must reside at 1,1. And that is the center of our circle. From there I'm going to drop an altitude down, from the base of triangle 2 down to the extended base of triangle 1. Because it's an altitude I know that that must be a right angle. Now again, if I look at the base of triangle 1 I can tell that this distance right here, this radius must be 4 because the distance between the x values is 4 and the y values don't change. So if I know that that radius is 4, all radii are congruent. So I know that this radius must be 4. And if I look at this altitude that I've dropped I know that it drops down to y value 1 and the y value of this point is 2 root 3 plus 1. Which means that it must have changed 2 times the square root of 3 units. So now I know the lengths of two sides of a right triangle which means that I can use a trig function to solve for any angle within the right triangle. And of course which angle am I going to solve for, theta the one that I want to know. So I would say that the sign of theta is equal to 2 root 3 over 4, which is the opposite side over the hypotenuse. I'm going to cancel out this 2 and I'm left with the sign equals root 3 over 2. And that equals approximately 0.866. And from there I'm going to turn right back around and plug that number into an inverse sign function, sign negative 1 of 0.866 which will give me 60 degrees. So I know that theta must equal 60 degrees. I'm Drew Moyer and that is how to calculate rotation in geometry.


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