In this assignment we will use the result of the assignment Inscribed Semicircle to find a geometric procedure to construct the square root of a number. We will illustrate this by constructing the square root of 3.
Problem 1. Draw a line 4 inches long, and mark points 1 inch and 2 inches from the left edge of the line. Draw a semicircle of radius 2 inches centered at the 2 inch mark, and then draw a vertical line starting at the 1 inch point and going up to the semicircle. Finally, inscribe a triangle in the semicircle by making the top corner of the triangle at the point where the vertical line hits the semicircle. The vertical line breaks this triangle into two smaller right triangles. By using what you did in the assignment Inscribed Semicircle, show that these are similar triangles, and explain why this yields the equation
Use this equation to show that r is the square root of 3.
Problem 2. Explain how to use the idea of the previous problem to construct the square root of any positive number.