About a Line and a Triangle

Date: 2015-10-07; view: 541.

Text 6. Make the written translation into Russian (time 90 minutes)

(1400 characters)

Given ∆ ABC, extend the side AB beyond the vertices. Now, rotate the line AB around the vertex A until it falls on the side AC. Next rotate it (from its new position) around C until it falls on the side BC.

Lastly, rotate it around B till it takes up its erstwhile position.

It is virtually obvious that although the line now occupies exactly the same position as before, something has changed. After three rotations, the line turned around 180°. So, for example, the point A will now lie on a different side from B than before. We say that turning the line around the triangle changed its orientation.

It appears that the line occupies the same position but not quite: points on the line did not preserve their locations. However, since there are just two possible orientations of the line, we come up with an interesting question: what happens to the line after it turns around the triangle twice? Will it occupy its original position exactly (point-for-point)?

The answer is easily obtained from the following observation.

After the first rotation the line occupies the same position but with a different orientation. Let's turn the line into coordinate axis. In other words, let's choose the origin – point O, the unit of measurements, and the positive direction. If, after the rotation, the point originally at the distance x from O will be now located at the position b-x. Therefore, there exists one point on the line that does not move even after a single rotation. This is the fixed point of the transformation. The fixed point solves the equation x = b-x. The rotation of the line around the triangle is simply equivalent to the rotation of the line around that point through 180°.