Question

a quadrilateral in the plane has vertices $(1, 3)$, $(1, 1)$, $(2, 1)$ and $(2006, 2007)$. how many square units is the area of the quadrilateral?

Answer 1

To find the area of the quadrilateral, we split it into two triangles and a rectangle and calculate their areas separately. The area of the quadrilateral is 8014 square units.

Step-by-step explanation:

To find the area of the quadrilateral, we can split it into two triangles and a rectangle. Two of the vertices form a vertical line segment, giving us the length of the rectangle. The other two vertices form a horizontal line segment, giving us the width of the rectangle. The area can be found by adding the areas of the two triangles and the rectangle together.

Let's calculate the area step by step:

1. Calculate the dimensions of the rectangle: the length is the distance between the points (2, 1) and (2006, 2007) in the x-direction, which is 2004. The width is the distance between the points (1, 3) and (1, 1) in the y-direction, which is 2.
2. Calculate the area of the rectangle: Area = length × width = 2004 × 2 = 4008 square units.
3. Calculate the area of the triangles: Each triangle has a base of 2 units (the length of the rectangle) and a height of 2003 units (the difference in y-coordinates between the points (2, 1) and (2006, 2007)). The area of one triangle is (base × height) / 2 = (2 × 2003) / 2 = 2003 square units. Since we have two identical triangles, the total area of the triangles is 2 × 2003 = 4006 square units.
4. Add the areas of the rectangle and the triangles: Total area = 4008 + 4006 = 8014 square units.

The area of the quadrilateral is 8014 square units.