Assignment 5A: The problem:

Given a segment EF, construct a point G on EF so that EG/EF = 1/(sqrt 3) using Thales theorem (i.e., use figure 4.4 and related ideas in BG).

Step 1. Construct two segments a and b with ratio a/b = 1/(sqrt 3).

We recognize that sqrt 3 appears in the "30-60" right triangle that is one-half of an equilateral triangle.

So construct any equilateral triangle ABC and an altitude CD.

Then if s = AB, then AD = s/2 and CD = (s/2)sqrt 3, so AD/CD = 1/sqrt 3.

Step 2. Given a segment EF construct G so that EG/EF = 1/(sqrt 3).

We start with segment EF.

Then construct a new ray with endpoint E. Then use your compass to draw arcs with center E of radius AD and CD to construct points M and N on the rays. Thus we have a segment EN of length CD with a point M on this segment, with EM = AD. Thus ratio EM/EN = 1/sqrt 3.

Draw the line NF; then construct the line through M parallel to NF, intersecting EF in G. This solves the problem.

(Another method would be the rectangle method from Assignment 4, but use Thales for this exercise. This method is very important.)

Discussion: What to look for when setting up a problem like this. In a Thales figure like this, the whole point is that the any ratios among segments on line AB are the same as the corresponding ratios on line AC cut by lines parallel to BC. In other words, the lengths on side AB are all scaled by a common factor k.

Thus, given segments 1 and a, and also segment b, to construct a segment of length x = b/a. To construct x, we start with the obvious relationship x = b/a but rewrite it explicitly as an equation of ratios: x/1 = b/a. Then mark off a and b on one segment and mark off 1 on a second segment, draw a line and construct a parallel to get equal ratios. The new length cut by the parallel is x. The method is spelled out exactly in BG 4.2, pp. 52-53.