Answer: See the diagram below for the construction.
The green segment DE represents the length of
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Follow these steps to construct
- Draw a really long horizontal line. Plot point A anywhere on this line.
- Use your compass to measure segment r. The width of the compass is as wide as the segment is. Transfer this width to the line drawn in step 1. You'll have the non-pencil part of the compass placed at point A. Then you'll draw an arc (aka piece of a circle) to have that arc intersect the horizontal line. Mark this point as point B. We can see that r = length of segment AB.
- We'll follow the same idea as step 2. This time we'll be transferring segment s. After doing so, we'll have point C on the segment such that BC = s. Also, point C is to the right of A and B. Refer to the diagram.
- Use your compass and straightedge to locate the midpoint of A and C. Let's call this point D. Note: D is the same location as B only when r = s.
- Use your compass to draw a circle centered at D, and that circle goes through points A and C. This makes AC a diameter of the circle.
- In the process of making the midpoint D (step 4), you should have constructed the perpendicular bisector. So you should have a vertical line through point D. This vertical line intersects the circle at point E. The length of segment DE is exactly
units long. The proof of this is for another time since the proof is fairly lengthy. For more info, you can search out "geometric mean theorem" with quotes.
Note: your drawing may have pencil arc markings that aren't shown in the diagram I made. Ideally these lines are faint not to distract from the darker segments.