Jiuzhang Suanshu (Gougu 15)

Suppose there is a right triangle whose gou (shorter leg) is 5 bu, and gu (longer leg) is 12 bu. What is the width of the inscribed square?

【Solution】

Let  be the shorter leg (勾 gou),  be the longer leg (股 gu), and  be the width of the inscribed square. Dissect the right triangle into 3 pieces (Figure 1 left): 1 red triangle, 1 blue triangle, and 1 yellow square.

Copy each shape two times (Figure 1 middle) and then assemble them to form one large rectangle (Figure 1 right). Use this rectangle to derive the formula for calculating the width of the inscribed square.

It is known that and , thus

Therefore, the width of the inscribed square is bu.