STRAIGHT-LINE AND COMPASS CONSTRUCTIONS.

Problem 13

Suppose you're shown a line segment and told that it is (exactly) one inch long.

13. Given a line segment of length z, show how you can construct a line segment of length (1/z).

Solution

1. This is a simple trick: the previous answer's construction is done 'backwards'.
2. Draw a right-angled triangle adg with d the right-angle such that ad = z and dg = 1.
3. Find the point, b on line ad (possibly extended through d) such that ab = 1.
4. Construct the normal to ab, passing through b. Where this intersects with ag, call the point of intersection c.
5. By similar triangles, bc / dg = ab / ad; in other words, bc / 1 = 1 / z.
6. The line segment bc is therefor of the desired length.