Question

Area of polygon having sides \[y-x-1=0,\text{ }2y-2x+2=0,\text{ }y-2x-2=0,\text{ }and\text{ }3y-6x-9=0\].
 (a) \[\dfrac{1}{4}\]
(b) \[\dfrac{1}{2}\]
(c) \[4\]
(d) \[2\]

Answer 1

Hint: The area of the parallelogram whose sides are:
\[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0,\text{ }{{a}_{1}}x+{{b}_{1}}y+{{d}_{1}}=0,\text{ }{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0\text{ }and\text{ }{{a}_{2}}x+{{b}_{2}}y+{{d}_{2}}=0\] is given as:
Area = \[\dfrac{\left| {{c}_{1}}-{{d}_{1}} \right|\left| {{c}_{2}}-{{d}_{2}} \right|}{\left| {{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}} \right|}\]. Apply this formula accordingly to find the area of the given polygon.

Complete step-by-step answer:
Given, the four sides of the polygon are:
\[y-x-1=0,\text{ }2y-2x+2=0,\text{ }y-2x-2=0,\text{ }and\text{ }3y-6x-9=0\]
From the given four equations of sides we can observe that the lines are \[y-x-1=0,\text{ }2y-2x+2=0\] parallel to each other because both the lines have the same slope of “1”.
Similarly, we can say that the lines \[y-2x-2=0\] and \[3y-6x-9=0\] are also parallel to each other since they have the same slope of “2”.
Therefore, the given polygon is a parallelogram.
A parallelogram is a closed shape four sided rectilinear figure with its opposite sides parallel to each other.
The area of the parallelogram whose sides are:
\[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0,\text{ }{{a}_{1}}x+{{b}_{1}}y+{{d}_{1}}=0,\text{ }{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0\text{ }and\text{ }{{a}_{2}}x+{{b}_{2}}y+{{d}_{2}}=0\]
 is given as:
Area = \[\dfrac{\left| {{c}_{1}}-{{d}_{1}} \right|\left| {{c}_{2}}-{{d}_{2}} \right|}{\left| {{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}} \right|}\]
Now, Let us consider the following:
\[y-x-1=0\] as \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0\] ,
\[y-x+1=0\] as \[{{a}_{1}}x+{{b}_{1}}y+{{d}_{1}}=0\],
\[y-2x-2=0\] as \[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0\], and
\[y-2x-3=0\] as \[{{a}_{2}}x+{{b}_{2}}y+{{d}_{2}}=0\].
Then, we shall have the following respectively:
\[{{a}_{1}}=-1,\text{ }{{b}_{1}}=1,\text{ }{{c}_{1}}=-1,\text{ }{{d}_{1}}=1\text{ }and,\text{ }{{a}_{2}}=-2,\text{ }{{b}_{2}}=1,\text{ }{{c}_{2}}=-2,\text{ }{{d}_{2}}=-3\]
Now, putting these values in the area of the parallelogram formula, we have:
Area = \[\dfrac{\left| -1-1 \right|\left| -2+3 \right|}{\left| -1+2 \right|}\]
          =\[\dfrac{\left( 2 \right)\left( 1 \right)}{\left( 1 \right)}\]
          = 2 square Units
So, the area of a given polygon is 2 square Units.
Hence, the correct answer is option d for this question.

Answer is option (d)

Note: The area of a polygon can also be found by finding out the vertices of a polygon and then applying the matrix method, but that would be lengthy and time consuming. Also note that we have to apply modulus in the area formula, since the area of a polygon cannot take negative values.