STRAIGHT-LINE AND COMPASS CONSTRUCTIONS.

Problem 10

Suppose you're shown a line segment and told that it is (exactly) one inch long.

10. Show how you can construct a line segment sqrt(2) inches long (hint: pythagoras).

Solution

1. Draw a line segment ab one inch in length.
2. Extend ab and draw the normal to ab passing though b. (Solution 2.)
3. Mark a point 1 inch along this new line from b. Call it c.
4. Construct the triangle abc. This is right-angled, with sides adjacent to the right-angle of length ab = 1 inch and cb = 1 inch.
5. From Pythagoras' theorem, ab squared + bc squared = ac squared. So the length ac is Sqrt(2) as required.