Given two line segments of lengths a and b, construct a line
segment with length of _{}.

[Problem submitted by Walter O’Connell, LACC Professor of Physics.]

**Solution for Problem 8: **

** **

Draw a circle with a diameter of length a + b. Then draw a
perpendicular to the diameter at the point that divide the diameter into a and
b. The angle opposite to the diameter is a right angle. This implies the two
smaller triangles are similar._{}

Note: Constructing a square that has the same area, as a given circle is one of the famous ancient problems in geometry. Professor O’Connell pointed out the above construction can be used to solve the problem as the following.

The given circle in the figure above is tangent to the
straight line at Q. _{}is a diameter. Roll the circle until P touches the line at M
(This operation is not allowed in classical geometry.) Then the length of _{} is _{}r. Construct a line segment with the length of_{}. Then use this line segment to construct a square.