# Chiu Chang Suann Jing/Examples/Example 8

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## Example of Problem from

## 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?*

## Solution

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

## Proof

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.

Hence:

\(\ds \paren {10 - x}^2\) | \(=\) | \(\ds x^2 + 3^2\) | 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 |

$\blacksquare$

## Historical Note

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

## Sources

- c. 100: Anonymous:
*Chiu Chang Suann Jing* - 1965: Henrietta Midonick:
*The Treasury of Mathematics: Volume $\text { 1 }$* - 1992: David Wells:
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