Identifying translations and rotations

• Feb 11th 2010, 07:08 PM
Pinkk
Identifying translations and rotations
Given the following diagram (note that the intersection point of the segment that connects triangles XYB and ABC on the segment AB should be point I), where the triangles are all equilateral, and the points H, I, and J bisect the segments AC, AB, and XB, respectively:

http://img651.imageshack.us/img651/9529/triangle.jpg

Identity the following translations and rotations:

* $\displaystyle R_{A, \frac{\pi}{2}} \circ R_{B, \frac{\pi}{2}}$
* $\displaystyle R_{C, \frac{\pi}{2}} \circ R_{A, \frac{\pi}{2}}$
* $\displaystyle R_{A, \frac{\pi}{2}} \circ T_{CA}$
* $\displaystyle R_{A, \frac{\pi}{3}} \circ R_{B, \frac{\pi}{3}}$
* $\displaystyle R_{A, \frac{\pi}{2}} \circ R_{B, \frac{\pi}{6}}$
* $\displaystyle R_{A, \frac{3\pi}{4}} \circ R_{B, \frac{\pi}{4}}$

I am absolutely confused and lost in the modern geometry class and have no idea how to approach these.

Edit: Is $\displaystyle R_{A, \frac{\pi}{2}} \circ R_{B, \frac{\pi}{2}} = R_{J, \frac{4\pi}{3}}$ ?
• Feb 12th 2010, 03:40 AM
HallsofIvy
Quote:

Originally Posted by Pinkk
Given the following diagram (note that the intersection point of the segment that connects triangles XYB and ABC on the segment AB should be point I), where the triangles are all equilateral, and the points H, I, and J bisect the segments AC, AB, and XB, respectively:

http://img651.imageshack.us/img651/9529/triangle.jpg

Identity the following translations and rotations:

* $\displaystyle R_{A, \frac{\pi}{2}} \circ R_{B, \frac{\pi}{2}}$
* $\displaystyle R_{C, \frac{\pi}{2}} \circ R_{A, \frac{\pi}{2}}$
* $\displaystyle R_{A, \frac{\pi}{2}} \circ T_{CA}$
* $\displaystyle R_{A, \frac{\pi}{3}} \circ R_{B, \frac{\pi}{3}}$
* $\displaystyle R_{A, \frac{\pi}{2}} \circ R_{B, \frac{\pi}{6}}$
* $\displaystyle R_{A, \frac{3\pi}{4}} \circ R_{B, \frac{\pi}{4}}$

I am absolutely confused and lost in the modern geometry class and have no idea how to approach these.

Edit: Is $\displaystyle R_{A, \frac{\pi}{2}} \circ R_{B, \frac{\pi}{2}} = R_{J, \frac{4\pi}{3}}$ ?

No, why in the world would you think so? You cannot possible get a rotation of $\displaystyle 4\pi/3$ by 2 rotations of $\displaystyle \pi/2$. If you wrote around point B $\displaystyle \pi/2$ radians the vertical line BH becomes a horizontal line, extending what is originally XB. Then rotating around the new position of A sends C into the original position of B. I'm not sure what is meant by "identifying" these but $\displaystyle R_{A,\frac{\pi}{2}} \circ R_{B,\frac{\pi}{2}}$ is a rotation around B by $\displaystyle \pi$ followed by a translation from B to C.
• Feb 12th 2010, 05:34 AM
Pinkk
From the way the book seems to describe, all the rotations are around the original points A,B, and C, if even they have already been translated. And by identity, they seem to always be able to reduce it to only one translation or only one rotation.

But don't worry, I'm fairly certain I'm dropping this class so it doesn't matter.
• Feb 12th 2010, 06:44 AM
Pinkk
Proof of rotations
Nevermind.
• Feb 12th 2010, 06:48 AM
Pinkk
Proof regarding rotations/reflections
Double posted by mistake.