Ruler and Compass Construction 15
Rectangle with Given Side Lengths

1. Extend ray AB
2. Semi-circle centered at B, radius CD
3. Arc centered at E, arbitrary radius
4. Arc centered at F, same radius; intersection G
5. Ray BG, which is perpendicular to AB
6. Intersection at H, which will be a corner of the rectangle
7. Arc centered at H, radius AB
8. Arc centered at A, radius CD
9. The rectangle ABHI
Why it works: A rectangle has right angles, so the construction does a perpendicular to AB at point B, conveniently using radius CD. Then it completes the rectangle by constructing the fourth point at I with equal opposite sides, which makes a parallelogram, which must be a rectangle since it already has a right angle at B.
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