1. First we find the mid point of AB. To do this without a ruler, put your compasses on one end, open them out to be somewhere near the other end of the line and draw a semicircle over the line AB. Repeat this at the other end of the line without altering the compass size. The two points where the semicircles cross can then be joined and this new line will cross AB at its mid point.
Now we are going to draw a line half the length of AB at point B, but at right-angles to the original line. This is where you use the set-square (but you CAN do this just using your compasses too - how?). So first draw a line at right angles to AB at end B.
Put your compasses on B, open them to the midpoint of AB and draw an arc to find the point on your new line which is half as long as AB. Now you have a new line at right angles to the original and exactly half as long as the original line.
Join the point just found to the other end of the original line (A) to make a triangle. Putting the compass point at the top point of the triangle and opening it out to point B (so it has a radius along the right-angle line) mark out a point on the diagonal which will also be half as long as the original line.
Finally, put the compass point at point A, open it out to the new point just found on the diagonal and mark a point the same distance along the original line. This point is now divides the original line AB into two parts, where the longer part AG is phi (0 61803..) times as long as the original line AB.
Was this article helpful?