Answer:
Area: 54 square units
Step-by-step explanation:
To find the area of the geometric shape defined by the given vertices without finding the lengths of the sides, we can use the Shoelace Formula. The Shoelace Formula allows you to calculate the area of a polygon when the coordinates of its vertices are known.
The formula is as follows:
[tex]\Large\boxed{\boxed{ \textsf{Area} = \dfrac{1}{2} \left| x_1y_2 + x_2y_3 + \ldots + x_ny_1 - y_1x_2 - y_2x_3 - \ldots - y_nx_1 \right|}} [/tex]
In this formula, [tex](x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)[/tex] are the coordinates of the vertices in counterclockwise order.
For the given vertices (0,0), (10,2), (13,8), and (3,6), the Shoelace Formula becomes:
[tex] \textsf{Area} = \dfrac{1}{2} \left| (0 \times 2 + 10 \times 8 + 13 \times 6 + 3 \times 0) - \\ (0 \times 10 + 2 \times 13 + 8 \times 3 + 6 \times 0) \right| [/tex]
Simplify the expression inside the absolute value:
[tex] \textsf{Area} = \dfrac{1}{2} \left| (0 + 80 + 78 + 0) - (0 + 26 + 24 + 0) \right| [/tex]
[tex] \textsf{Area} = \dfrac{1}{2} \left| 158 - 50 \right| [/tex]
[tex] \textsf{Area} = \dfrac{1}{2} \times 108[/tex]
[tex] \textsf{Area} = 54 [/tex]
So, the area of the geometric shape is 54 square units.
