Chiu Chang Suann Jing/Examples/Example 6

Example of Problem from Chiu Chang Suann Jing

There is a pool $10$ feet square, with a reed growing vertically in the centre,

its roots at the bottom of the pool, which rises $1$ foot above the surface;

when drawn towards the shore it reaches exactly to the brink of the pool;

what is the depth of the water?

The water is $12$ feet deep.

Let the depth of the water be $d$.

The length of the reed is $d + 1$.

When drawn to the edge of the pool, the reed forms the hypotenuse of a right triangle.

One of the legs of that right triangle is the depth of the pool, which is $d$.

The other leg is the distance from the centre of the pool, which is $5$ feet.

Hence:

$\ds \paren {d + 1}^2$

$=$

$\ds d^2 + 5^2$

$\ds \leadsto \ \ $

$\ds 2 d + 1$

$=$

$\ds 25$

simplification

$\ds \leadsto \ \ $

$\ds d$

$=$

$\ds 12$

simplification

$\blacksquare$