Question

Kamala has a triangular field with sides $240m,200m,360m,$ where she grew wheat. In another triangular field with sides $240m,320m,400m,$ adjacent to the previous field, she wanted to grow potatoes and onions. She divided the field in two parts by joining the mid-point of the longest side to the opposite vertex and grew potatoes in one part and onions in the other part. If $m$ is the area ( in hectares ) has been used for potatoes? (1 hectare = $10000m^2$).Find $100m$

Answer 1

$E$ is the midpoint of $AD$, then

$AE=ED=200m$

Area of wheat field $=ar△ABC$

Area of potato field $=ar△AEC$

Area of onion field $=ar△EDC$

$ar△ABC$ $=\sqrt{S(S-a)(S-b)(S-c)},S=\frac{200+240+360}{2}=400m=\sqrt{400(400-200)(400-240)(400-360)}=\sqrt{400\times 200\times 160\times 40}=16000\sqrt{2}{m}^2=1.6\sqrt{2}\text{hectare}=1.6\times 1.414=2.26\text{hectares}$

In $△ACD$

$△AEC$ and $△CED$ have same base $[arAE=ED=200m]$

and they have same height $CF$

Therefore, $ar△AEC=ar△CED=\frac{1}{2}ar△ACD$

$ar△ACD$ $=\sqrt{S(S-a)(S-b)(S-c)},S=\frac{240+320+400}{2}=480m=\sqrt{480(480-240)(480-320)(480-400)}=\sqrt{480\times 240\times 160\times 80}=38400m^2=3.84hectares$

$\therefore ar△AEC=ar△CED=\frac{1}{2}\times 3.84=1.92\text{hectares}$

Therefore,

Area of wheat field $=2.26\text{hectares}$

Area of potato field $=1.92\text{hectares}$

Area of onion field $=1.92\text{hectares}$

For potato $100m=100\times 1.92=192\text{hectares}$