STRAIGHT-LINE AND COMPASS CONSTRUCTIONS.

Problem 12

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

12. Given two additional line segments, of lengths x inches and y inches (you don't necessarily know x and y; they can be arbitrary lengths), can you construct a line segment of length (x times y)?

Solution

1. Construct a right-angled triangle abc such that b is the right-angle, ab is of length 1 and bc is the same length as x. The process for this was given as part of the solution to question 10.
2. Extend ab. Mark a point, d, on the extension of ab such that ad has length y.
3. Construct the normal to ad passing through d. Call this line E
4. Extend the line ac. Call this line F. Let the intersection of E and F be g.
5. Note that abc and adg are similar triangles; therefore, their corresponding sides will be in the same ratio.
6. Thus, ad / ab = dg / bc. In other words, y / 1 = dg / x. Multiplying both sides by x, we get the result dg is of length x times y, as required.