Chiu Chang Suann Jing/Examples/Example 8

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Example of Problem from Chiu Chang Suann Jing

There is a bamboo $10$ feet high,
the upper end of which being broken down on reaching the ground,
the tip is just $3$ feet from the stem;
what is the height of the break?


The height of the break is $4 \tfrac {11} {20}$ feet.


Let the height of the break be $x$ feet.

The length of the broken section is then $10 - x$ feet.

The broken section forms the hypotenuse of a right triangle.

One of the legs of that right triangle is the remaining stalk of bamboo, which is $x$ feet long.

The other leg is the distance of the end of the tip from the stalk, which is $3$ feet.


\(\ds \paren {10 - x}^2\) \(=\) \(\ds x^2 + 3^2\) Pythagoras's Theorem
\(\ds \leadsto \ \ \) \(\ds 20 x\) \(=\) \(\ds 100 - 9\) simplification
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds \frac {91} {20}\) simplification
\(\ds \) \(=\) \(\ds 4 \tfrac {11} {20}\) simplification


Historical Note

This problem was also presented by Brahmagupta in one of his works, but it has not been identified where this appears.