Construct the Square Root of 5

This construction uses the constructions of √2 and √3. The algebraic formula is (√2)2 + (√3)2 = (√5)2.

DiagramInstructions
1. Let line segment AB be unity (a line segment of length 1).
2. Construct √2 using AB as unity. How?
3. Construct √3 using AB as unity. How?
4. Draw a line parallel to AB through point D.
5. Draw a line of length AC with endpoint D in the parallel line.
6. Mark the other endpoint of the line just drawn as E.
7. Draw line segment C'E. The length of C'E is √5.

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 C'E to AB is always √5.

Proof

  1. The length of AB is taken to be 1 by definition.
  2. By construction, the length of AC is √2.
  3. By construction, the length of C'D is √3.
  4. Since DE is constructed to be the same length as AC, the length of DE is √2.
  5. By the Pythagorean Theorem (see Euclid's Proposition 47), DE2 + C'D2 = C'E2.
  6. By substitution, we get (√2)2 + (√3)2 = C'E2.
  7. By simplifying, we get 2 + 3 = C'E2.
  8. 5 = C'E2.
  9. √5 = C'E

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