We have seen how to get 90° using paper folding. Now, suppose we do not have any paper but two sticks of equal length, and a thread. How do we make an exact 90° using these?
Step 1: Place the two sticks of equal length such that they cross at their midpoints.
Step 2: Pass a thread through the four endpoints of the sticks.
Result: The thread will form a rectangle with all angles 90°.
Why it works: Since the sticks (diagonals) are equal and bisect each other, the thread forms a rectangle. When diagonals are equal and bisect each other, all angles of the quadrilateral are 90°. The angle between adjacent sides of this rectangle gives the exact 90°.
Explanation
The chapter's Carpenter's Problem demonstrates this exact method. When two equal sticks (diagonals) are joined at their midpoints, and a thread is passed through their endpoints, a rectangle is formed. As established in the chapter, when diagonals are equal and bisect each other, all angles of the quadrilateral are 90°. This method is used by carpenters in Europe and farmers in Mozambique to create rectangular frames.
Solution Steps
Place the two sticks of equal length such that they cross at their midpoints.
Pass a thread through the four endpoints of the sticks.
The thread will form a rectangle with all angles 90°.
Use the angle between adjacent sides of this rectangle as the exact 90°.