Lesson Plan
Given a line segment AB, describe how to construct a segment whose length is times the length of AB, using a compass and unmarked straightedge.
1. Create a line segment with unit length 1
2. Make another line segment twice the unit length perpendicular to the original length
3. Connect to make a right triangle
4. Create an altitude from the hypotenuse to the right angle, label it x
5. Solving for x using similar triangles gives us √2
6. Repeat steps 1 – 5 with segments of length 1 and 3
7. Solving for x gives us a segment with length √3
8. Move new segments from steps 5 and 7, taking off a length of 1 from each.
9. Connect segments from step 8 at one point O forming an arbitrary acute angle
10. Segment √2 –1 will be OP and segment √3 – 1 will be OQ
11. Extend the segment √2 –1 to have a length of 1,call it OR
12. Connect P and Q, make a parallel line to PQ going through R
13. Extend the segment √3 –1 to intersect with parallel line going through R, call point of intersection S
14. Connect point O with S, this is line segment x
15. Now you can create a ratio with