An acute angle ∠ABC and interior ray BD are given, as shown. Laura is given an infinite ruler which consists of two parallel rays joined on one end by a segment perpendicular to both of them. One may place the infinite ruler onto the diagram so that one of the infinite edges (marked with an arrow) passes through any two selected points in the diagram, or so that any edge of the ruler coincides with (i.e. exactly overlaps with) a portion of a segment, ray or line already in the diagram. Once the ruler is placed, one may draw any edge of the ruler "onto the diagram", in the usual fashion. One may also plot points where any two straight objects intersect.
Using only an infinite ruler, describe how to construct points P on ray BA and Q on ray BC such that ray BD intersects ray PQ at a point R with PR=2(QR). Then prove that your construction works.
Solution
Solution:
Place the infinite ruler so that one infinite edge aligns with ray BD and the other infinite edge intersects ray BC, then trace its outline. Do the same thing twice on the other side of ray BD, so that the final edge intersects ray BA. These points of intersection give the desired segment PQ.
We outline the proof, which is mainly self-evident. The construction produces a set of four equally spaced rays, which therefore cut PQ into three congruent segments. It follows that PR=2(QR).
Want a route through all this instead of an archive? The track
puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.
Source: MathNet,
licensed CC-BY-4.0.
Statement reproduced verbatim; metadata (topic, difficulty) added by this project.
