Question

Anita has triangular field of sides 200m 360m, and 240m where she grew wheat. In other triangular field with sides 400m, 240m and 320m adjacent to the previous field, she wanted to grow potatoes and tomatoes. 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 tomato in other part. How much area (in hectors) has been used for wheat, potatoes, and tomatoes ?​

Answer 1

Answer:

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 =

S(S−a)(S−b)(S−c)

,S=

2

200+240+360

=400m

=

400(400−200)(400−240)(400−360)

=

400×200×160×40

=16000

2

m

2

=1.6

2

hectare

=1.6×1.414=2.26hectares

In △ACD

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

and they have same height CF

Therefore, ar△AEC=ar△CED=

2

1

ar△ACD

ar△ACD =

S(S−a)(S−b)(S−c)

,S=

2

240+320+400

=480m

=

480(480−240)(480−320)(480−400)

=

480×240×160×80

=38400m

2

=3.84hectares

∴ar△AEC=ar△CED=

2

1

×3.84=1.92hectares

Therefore,

Area of wheat field =2.26hectares

Area of potato field =1.92hectares

Area of onion field =1.92hectares

For potato 100m=100×1.92=192hectares

Answer 2

Given:

Sides of Wheat Field, a = 200

                                    b = 360,

                                    c = 240

Sides of the Other Field, a1 = 400,

                                         b1 = 320,

                                           c = 240

To Find:

The Area that is used for Wheat, Potatoes, and Tomatoes(In Hectares).

Solution:

For Finding the Area of Wheat Field,

By using Heron's formula,

                              $Area = \sqrt{s((s-a)(s-b)(s-c))}$

Here, '$s$' is the Semi-Perimeter,

                                    $s=\frac{a+b+c}{2}$

                                    $s= \frac{200+360+240}{2}$

                                    $s = \frac{800}{2}$

                                    $s= 400$

And, a,b, c are the sides of the triangle.

So,

                               $Area = \sqrt{400((400-200)(400-360)(400-240))}\\Area =\sqrt{ 400(200*40*160)}\\Area = \sqrt{400(1280000)}\\Area = \sqrt{512000000}\\Area = 22,627.41$

Hence the Area of Wheat Field is 22,627.41 m²

For Finding the Area of Potato and Tomato Fields,

Now, firstly the total area of the other field.

By using same Heron's Formula,

                                     $s=\frac{a1+b1+c}{2}$

                                     $s= \frac{400+320+240}{2}$

                                     $s = \frac{960}{2}$

                                     $s= 480$

Now the Total Area is,

                               $Area = \sqrt{480((480-400)(480-320)(480-240))}\\Area =\sqrt{ 480(80*160*240)}\\Area = \sqrt{480(3072000)}\\Area = \sqrt{1474560000}\\Area = 38400$

Since the Field is divided by the line from the mid-point of the longest side to the vertex. This line is the median of that triangular field which divides the field's area into two equal parts.

So the Area for Tomato and Potato Fields will be 19200 m² each.

For Area to be in Hectares,

The Area should be divided by 10,000.

So,

Area of Wheat Field  $=\frac{22627.41}{10000}$

                                  $= 2.26$

∴ The Area of Wheat Field is 2.26 hectares.

Area of Tomato Field $=\frac{19200}{10000}$

                                   $=1.92$

∴ The Area of Tomato and Potato Field is 1.92 hectares each.