Question

Area of pentagon formed by $\left( 1,2 \right),\text{ }\left( 2,\text{ }-1 \right),\text{ }\left( -2,\text{ }-1 \right),\text{ }\left( 2,1 \right)\text{ }and\text{ }\left( -1,2 \right)$.
A) 20
B) 15
C) 4
D) 10

Answer 1

Hint: As the five coordinates of the pentagon are provided, you can find the area using the formula: Area of pentagon = \[\dfrac{1}{2}\left| ({{x}_{1}}{{y}_{2}}+{{x}_{2}}{{y}_{3}}+{{x}_{3}}{{y}_{4}}+{{x}_{4}}{{y}_{5}}+{{x}_{5}}{{y}_{1}})-({{y}_{1}}{{x}_{2}}+{{y}_{2}}{{x}_{3}}+{{y}_{3}}{{x}_{4}}+{{y}_{4}}{{x}_{5}}+{{y}_{5}}{{x}_{1}}) \right|\]. Substitute the coordinates accordingly.

Complete step-by-step answer:
A pentagon is any five sided polygon. The sum of the internal angles in a pentagon is equal to 540 degrees. A regular pentagon will have all the sides and angles as equal. Pentagons can be regular or irregular and convex or concave in nature.
Given, the five coordinates of the pentagon are:
$\left( 1,2 \right),\text{ }\left( 2,\text{ }-1 \right),\text{ }\left( -2,\text{ }-1 \right),\text{ }\left( 2,1 \right)\text{ }and\text{ }\left( -1,2 \right)$.
Now, to find the area of pentagon we can use the below formula accordingly:
Area of pentagon =
\[\dfrac{1}{2}\left| ({{x}_{1}}{{y}_{2}}+{{x}_{2}}{{y}_{3}}+{{x}_{3}}{{y}_{4}}+{{x}_{4}}{{y}_{5}}+{{x}_{5}}{{y}_{1}})-({{y}_{1}}{{x}_{2}}+{{y}_{2}}{{x}_{3}}+{{y}_{3}}{{x}_{4}}+{{y}_{4}}{{x}_{5}}+{{y}_{5}}{{x}_{1}}) \right|\]
Now, the given five coordinates can be assumed accordingly:
Where, the coordinate $\left( 1,2 \right)$ is assumed to be \[({{x}_{1}},{{y}_{1}})\].
Similarly, the rest of the coordinates can be assumed as:
The coordinate \[\left( {{x}_{2}},{{y}_{2}} \right)\text{ = }\left( 2,-1 \right)\]
The coordinate \[\left( {{x}_{3}},{{y}_{3}} \right)\text{ = }\left( -2,-1 \right)\]
The coordinate \[\left( {{x}_{4}},{{y}_{4}} \right)\text{ = }\left( 2,1 \right)\]
The coordinate \[\left( {{x}_{5}},{{y}_{5}} \right)\text{ = }\left( -1,2 \right)\]
Now, let us substitute the coordinates carefully in the formula, following the order given, to find the area of our given polygon:
Therefore, the area will be equal to:
Area = \[\dfrac{1}{2}\left| ((-1)+(-2)+(-2)+(4)+(-2))-((4)+(2)+(-2)+(-1)+(2)) \right|\]
         = \[\dfrac{1}{2}\left| (-3)-(5) \right|\]
         =\[\dfrac{1}{2}\left| -8 \right|\]
         = 4 square units
Area = 4 square units.
So, the area of the given polygon is 4 square units.
Hence, option (C) is the correct answer.
Note: We need to consider the modulus in the formula, as the area of the pentagon cannot take a negative value. Also, follow the order that is mentioned in the formula properly to avoid a wrong answer.