[Geometry] Quadrature of the Triangle

2021-10-19 16:22 作者:AoiSTZ23 0人读过 | 我要投稿

By: Tao Steven Zheng (郑涛)

【Problem】

Shushu Jiuzhang (Tianyulei 2)
There is a field of sand with three sides. The short side is 13 ''li'', the medium side is 14 ''li'', and the long side is 15 ''li''. The ''li'' rule is 300 ''bu''. How large is the field?

【Solution】

In the ''Shushu Jiuzhang'', Qin Jiushao gave the formula:

$A%20%3D%20%5Csqrt%7B%5Cfrac%7B1%7D%7B4%7D%20%5Cleft%5Ba%5E2%20c%5E2%20-%20%7B%5Cleft(%5Cfrac%7Ba%5E2%2Bc%5E2-b%5E2%7D%7B2%7D%20%5Cright)%7D%5E%7B2%7D%20%5Cright%5D%7D%20$

where $A$ is the area of a triangle, $a$ is the short side, $b$ is the middle side, and $c$ is the long side (derivation below).

$A%5E2%20%3D%20%5Cfrac%7B1%7D%7B4%7D%20%5Cleft%5Ba%5E2%20c%5E2%20-%20%7B%5Cleft(%5Cfrac%7Ba%5E2%2Bc%5E2-b%5E2%7D%7B2%7D%20%5Cright)%7D%5E%7B2%7D%20%5Cright%5D$

$A%5E2%20-%207056%20%3D%200$

$A%20%3D%2084$

The area Qin Jiushao gave is 315 ''qing''. This is calculated by knowing that 1 ''li'' is 300 ''bu'', 1 ''mu'' is 240 ''sq.bu'', and 1 ''qing'' is 100 ''mu''.

$84%20%5C%3Bsq.li%20%5Cleft(%5Cfrac%7B300%20%5C%3B%20bu%7D%7B1%20%5C%3Bli%7D%20%5Cright)%20%5Cleft(%5Cfrac%7B300%20%5C%3B%20bu%7D%7B1%20%5C%3Bli%7D%20%5Cright)%20%5Cleft(%5Cfrac%7B1%20%5C%3B%20mu%7D%7B240%20%5C%3B%20sq.bu%7D%20%5Cright)%20%5Cleft(%5Cfrac%7B1%20%5C%3B%20qing%7D%7B100%20%5C%3B%20mu%7D%20%5Cright)%20%3D%20315%20%5C%3B%20qing$

【Derivation】

Let $c$ be the base, $h$ be the height, and $A$ be the area of the triangle, then

$A%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20ch%20$

From the above diagram, determine two equations for $h%5E2$ :

Equation 1
$h%5E2%20%3D%20a%5E2%20-%20d%5E2$

Equation 2
$h%5E2%20%3D%20b%5E2%20-%20%7B(c-d)%7D%5E%7B2%7D$

Equate both equations and solve for $d$ :

$a%5E2%20-%20d%5E2%20%3D%20b%5E2%20-%20(c%5E2%20-%202dc%20%2B%20d%5E2)$

$c%5E2%20%2B%20a%5E2%20-%20b%5E2%20%3D%202dc$

$d%20%3D%20%5Cfrac%7Bc%5E2%2Ba%5E2-b%5E2%7D%7B2c%7D$

Substitute this result into Equation 1 and take the positive square root:

$h%5E2%20%3D%20a%5E2%20-%20%7B%5Cleft(%5Cfrac%7Bc%5E2%2Ba%5E2-b%5E2%7D%7B2c%7D%5Cright)%7D%5E%7B2%7D$

$h%20%3D%20%5Csqrt%7Ba%5E2%20-%20%7B%5Cleft(%5Cfrac%7Bc%5E2%2Ba%5E2-b%5E2%7D%7B2c%7D%5Cright)%7D%5E%7B2%7D%7D$

The area of the triangle is therefore

$A%20%3D%20%5Cfrac%7B1%7D%7B2%7Dc%20%5Csqrt%7Ba%5E2%20-%20%7B%5Cleft(%5Cfrac%7Ba%5E2%2Bc%5E2-b%5E2%7D%7B2%7D%20%5Cright)%7D%5E%7B2%7D%7D$