2: CROP FIELD : A farmer has a 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.
1. Which shape is formed with the region of wheat
crop?
2. The farmer wants to fence the field with the wire,
and then what is the length of wire required if side of
rhombus is 12
Answers
Given:
ABCD is a rhombus.
Sode of the rhombus=12 units
To find:
i. The shape is formed with the region of the wheat crop
ii. The length of wire required to fence the field
Solution:
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.
ii. 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.
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.
Answer:
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