Construct the Square Root of 3

DiagramInstructions
Line segment AB is unity.1. Let the line segment AB be unity (a line segment of length 1).
Add a circle with center A and radius AB.2. Construct a circle with center A and radius AB
Add a circle with center B and radius AB.3. Construct a circle with center B and radius AB
Add points C and D at the intersection of the circles.4. Mark the intersections of the two circles C and D.
Draw the line segment CD. The length of the segment is square root of 3.5. Draw line segment CD. The length of line segment CD is √3.
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You can change the figure by clicking and dragging on points A and B. Notice that while the measure of the length of AB changes, the ratio of the length of CD to AB is always √3.

Proof

  1. The length of AB is taken to be 1 by definition.
  2. Because both circles are constructed with a radius of AB, they are congruent.
  3. Since the segments CA and CB are both radii of congruent circles, they are congruent.
  4. Since AB is perpendicular to CD (see Euclid's Proposition ??), ∠ CMB is a right angle.
  5. By the Pythagorean Theorem (see Euclid's Proposition 47), BM2 + CM2 = BC2.
  6. But, since CD is unity, CM = 1/2.
  7. Since CB = unity by construction, CB = 1.
  8. CM2 + (1/2)2 = 12.
  9. Since CM = 1/2 CD, (1/2 CD)2 + (1/2)2 = 12.
  10. Simplifying give us (1/2 CD)2 + 1/4 = 1.
  11. 1/4 CD2 = 3/4.
  12. CD2 = 3.
  13. CD = √3, QED.

Citation

Cite this article as:
McAdams, David, Construct the Square Root of 3, from LifeIsAStoryProblem.org, 30 June 2007, , URL https://lifeisastoryproblem.tripod.com/numbers/cons_sqrt_3.html.

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