Question

If $A(0,0)$, $B(12,0)$,$C(12,2)$,$D(6,7)$,$E(0,5)$ are the vertices of the pentagon $ABCDE$, then its area in square units, is
A. 58
B. 60
C. 61
D. 62
E. 63

Answer 1

Hint: In the question we are given the coordinates of all the 5 points. We will use the formula of area of pentagon which is
Area
$A = \dfrac{1}{2}\sum {{x_n}{y_{n + 1}} - {y_n}{x_{n + 1}}} $
 where n = 1, 2, 3, 4, 5. We substitute the values and find the area.

Formula Used:
$A = \dfrac{1}{2}\sum {{x_n}{y_{n + 1}} - {y_n}{x_{n + 1}}} $

Complete step by step Solution:
Pentagon is a figure with 5 straight sides and 5 angles. A regular pentagon has 5 lines of symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216°, and 288°).
A convex pentagon has no angles pointing inwards and no internal angles can be more than 180°. When any internal angle is greater than 180° it becomes a concave pentagon.
A convex pentagon is defined as one whose vertices where the sides meet, are pointing outwards unlike a concave pentagon whose vertices point inwards.
The given vertices are $A(0,0)$, $B(12,0)$,$C(12,2)$,$D(6,7)$,$E(0,5)$
Now let $({x_1},{y_1}) = (0,0)$
$({x_2},{y_2}) = (12,0)$
$({x_3},{y_3}) = (12,2)$
$({x_4},{y_4}) = (6,7)$
$({x_5},{y_5}) = (0,5)$
Now, substituting the value in formula of area of a pentagon we get,
$A = \dfrac{1}{2}[{x_1}{y_2} + {x_2}{y_3} + {x_3}{y_4} + {x_4}{y_5} + {x_5}{y_1} - ({y_1}{x_2} + {y_2}{x_3} + {y_3}{x_4} + {y_4}{x_5} + {y_5}{x_1})]$
$A = \dfrac{1}{2}[0 \times 0 + 12 \times 2 + 12 \times 7 + 6 \times 5 + 0 \times 0 - (0 + 0 + 2 \times 6 + 7 \times 0 + 0 \times 5)]$
$A = 63$
Hence, the area of the pentagon is 63 sq. units.

Hence, the correct option is E.

Note: The properties of a simple pentagon are that it must have five straight sides that meet to create five vertices but they should not self-intersect: Pentagons have 5 interior angles, which sum to 540°. The five sides do not intersect. The formula given can also be used for all polygons. There are also many ways to calculate the area of a pentagon like we can convert it into different shapes and calculate the area of each shape. It depends on the value given in the question we have to deal accordingly.