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A 10-ft-long stem of bamboo is broken in such a way that its tip touches the ground 3ft from the base of the stem. What is the height of the break? From the Chinese mathematics textbook "Chui-chang suan-shu", or "Nine Chapters on The Mathematical Art". 250 B.C.

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4.55 ft assuming the stem is perpendicular to the ground. Hypotenuse would be 5.45

 

The base of 3 is not part of the stem

 

5.45 sq = 4.55 sq + 3 sq

Assuming the stem is perpendicular to the ground, call the distance from the ground to the break h.  Then, the hypotenuse of the right triangle formed will have length 10 - h.  Using the Pythagorean theorem:

 

h^2 + 9 = (10 - h)^2

h^2 + 9 = 100 -20h + h^2  //canceling out the h^2 on each side gets

9 = 100 - 20h 

 

Solving the linear equation reveals h=4.55

 

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