Unizor – Creative Mind through Art of Mathematics: Geometry+ 10: UNIZOR.COM – Math+ & Problems

Notes to a video lecture on http://www.unizor.com

Geometry+ 10

Problem A

Given two parallel lines and a segment AB on one of them.
Using only a straight line ruler, increase the length of segment AB by a factor of N.
In other words, find a point B’ on the same line as AB such that the length of AB’ is N times greater than the length of AB.

Solution A
Let’s repeat the doubling of a segment, explained in the previous lecture Geometry 09 as Problem C.
Assume, AB is on the lower parallel line as on the picture below.

Choose any segment CD along the upper parallel line and divide it in halves by point P, as described in the Problem B of lecture Geometry 09.
Connect A and C, connect B and P. Lines AC and BP intersect at some pointM (if they don’t and happened to be parallel, choose a longer CD.)
Now connect M and D and extend it to intersect with the lower parallel line at point B’.
The segments AB and BB’ have equal length because segments CP and BP have equal length and two pairs of triangles are similar:
ΔAMB is similar to ΔCMP,
ΔBMB’ is similar to ΔPMD.
Now we are ready to increase the length of AB by any factor.
To do this, just repeat the doubling of the size for segment BB’ getting point B”, so segment AB” has a triple length of AB.
Repeating this procedure any number of times we will get the new segment’s length any number of times larger than the length of AB.

Problem B

Given two parallel lines and a segment AB on one of them.
Using only a straight line ruler, divide segment AB into N sub-segments of equal length.
In other words, find points B₁, B₂, …,B_N−1 on segment AB such that the length of any segment B_iB_i+1 is 1/N^th of the length of AB for any i∈[0,N−1], assuming A= B₀ and B=B_N.

Solution B
Assume, AB is on the lower parallel line.
Choose any segment CD along the upper parallel line and increase its length by a factor of N as described in the Problem A. Associate symbol D₀ with point C, D₁ with D and new points that double the length of a previous segment will be D₂, D₃, …,D_N.

Connect A=B₀ with C=D₀ and B with D_N, extending these two lines to an intersection point M (if they are parallel and do not intersect, choose different point D.)
Connecting point M with each D_i and extending to intersect with AB, gives all the points B_i.