# Chiu Chang Suann Jing/Examples/Example 8

## 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.