Question

7.
CROP FIELD
A farmer havea field ABCD. Field ABCD is in the shape of rhombus and P, Q, R and S are the mid-points
of the sides AB, BC, CD and DA respectively. The farmer wants to crop wheat in area PQRS and mustard
crop in rest of the region.
(a) Which shape is formed with the region of wheat crop?
(b) The farmer wants to fence the field with the wire, and then what is
the length of wire required if side of rhombus is 12m.
(c) Find the diagonal AC if the side of rhombus field 12m and diagonal
BD is 20m?
(d) What is the use of mustard crop in our daily life?
bloc ly tho tog​

Answer 1

Given:

ABCD is in the shape of rhombus

P, Q, R and S are the mid-points of the sides AB, BC, CD and DA

To Find:

a) Which shape is formed with the region of wheat crop

(b) The farmer wants to fence the field with the wire, and then what is

the length of wire required if side of rhombus is 12m.

(c) Find the diagonal AC if the side of rhombus field 12m and diagonal

BD is 20m

(d) What is the use of mustard crop in our daily life

SOLUTION:

a) shape formed with the region of wheat crop

Given that,

ABCD is in the shape of rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA

By the figure,

If ABCD is a rhombus and P, Q, R, S are mid-points of the sides AB, BC, CD, and DA respectively, then the quadrilateral PQRS is a rectangle.

It is given that, The farmer wants to crop wheat in area PQRS

Then, A rectangle shape is formed with the region of the wheat crop.

b)The length of wire required if the side of the rhombus is 15m.

To find the length of wire required, we calculate the perimeter of rhombus

given, side of rhombus = a = 12m

Perimeter of Rhombus = 4a

                                      = 4 ×12

                                      = 48 m

The length of wire required is 48 m

c) The diagonal AC

AC is perpendicular to BD

Given , one diagonal BD = 20m

      side of rhombus field= 12m

To Find another diagonal we use Pythagoras theorem which states that "In a right-angled triangle OAB,  the square of the hypotenuse side is equal to the sum of squares of the other two sides"

AB² = OA² + OB²

Here, Let O be the centre where the diagonals AC & BD intersect

If BD = 20 then BD = OB + OD & OB is half of BD that is OB = 10m

AB = 12m

AB² = OA² + OB²

12² = OA² + 10²

OA ² = 144 -100

OA² = 44

0A  = $\sqrt{11\times 4}$

OA = 2 $\sqrt{11}$

AC = 2 $\sqrt{11}$+2 $\sqrt{11}$

AC = 4 $\sqrt{11}$

The diagonal AC is 4 $\sqrt{11}$ m

d) The use of mustard crop in our daily life

• Mustard seeds and its oil is used for culinary purpose

• Its younger leaves are often used in vegetable dishes
• Mustard oil cake is another crushed product other than mustard oil, which finds use in feeding cattle

Answer 2

Given:

ABCD is a rhombus.

Side of the rhombus=12 units

To find:

a. The shape is formed with the region of the wheat crop

b. The length of wire required to fence the field

(c) Find the diagonal AC

(d) The uses of the mustard crop in our daily life

Solution:

a) The shape formed with the region of the wheat crop is a rectangle.

The length of the wire required to fence the field is 48 units.

We will find the solution by following the given steps-

i. We know that the field is in the shape of a rhombus.

In rhombus ABCD, the diagonal AC divides the rhombus into two triangles, ∆ABC and ∆ADC.

In ∆ABC, P and Q are the midpoints of AB and BC.

As per the midpoint theorem, PQ is parallel to AC and PQ=1/×AC

Similarly, in ∆ADC, S and R are the midpoints of AD and DC.

As per the midpoint theorem, SR is parallel to AC and SR=1/2×AC

So, PQ=SR

By following a similar pattern, PS=QR=1/2×BD.

Now we will join PR and QS.

We get PBCR and SABQ are parallelograms.

So, PR=BC and SQ=AB

Since ABCD is a rhombus, all its sides are equal, and thus, AB=BC.

Also, PR=SQ.

In figure PQRS, PQ=SR, PS=QR, and PR=SQ.

The opposite sides are equal and the diagonals are equal as well.

Therefore, PQRS is a rectangle.

b). The fencing of the field would require covering the perimeter of the rhombus-shaped field.

We know that in rhombus ABCD, AB=BC=CD=AD.

The side of the rhombus=12 units.

Perimeter= AB+BC+CD+AD

=12+12+12+12

=48 units

The length of the wire for fencing=48 units.

c) The diagonal AC

AC is perpendicular to BD

Given , one diagonal BD = 20m

side of rhombus field= 12m

To Find another diagonal we use Pythagoras theorem which states that "In a right-angled triangle OAB,  the square of the hypotenuse side is equal to the sum of squares of the other two sides"

AB² = OA² + OB²

Here, Let O be the centre where the diagonals AC & BD intersect

If BD = 20 then BD = OB + OD & OB is half of BD that is OB = 10m

AB = 12m

AB² = OA² + OB²

12² = OA² + 10²

OA ² = 144 -100

OA² = 44

0A  = $\sqrt{4X11$

OA = 2 $\sqrt{11}$

AC = 2 $\sqrt{11}$+2 $\sqrt{11}$

AC = 4 $\sqrt{11}$

The diagonal AC is 4 m

d) The use of the mustard crop in our daily life

• Mustard seeds and their oil is used for culinary purpose
• Its younger leaves are often used in vegetable dishes
• Mustard oil cake is another crushed product other than mustard oil, which finds use in feeding cattle

Therefore, the shape formed with the region of the wheat crop is a rectangle, and the length of the wire required to fence the field is 48 units.