The image shows a graph with a quadrilateral $RSTU$ and the task is to find its area. The vertices of the quadrilateral are as follows:

• $R(-4, -1)$
• $S(-4, 4)$
• $T(3, 3)$
• $U(2, -1)$

To find the area of the quadrilateral with vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4)$, we can use the shoelace formula (also known as the Surveyor's Formula):

$\text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right|$

Let’s apply this formula to the given vertices:

• $R(-4, -1)$
• $S(-4, 4)$
• $T(3, 3)$
• $U(2, -1)$

I will now calculate the area.The area of quadrilateral $RSTU$ is $29.5$ square units.

Would you like a detailed explanation of the steps involved in applying the shoelace formula, or do you have any other questions?

Here are some related questions to explore:

1. How does the shoelace formula work for polygons with more vertices?
2. Can the shoelace formula be used for polygons with holes or complex shapes?
3. What are other methods to find the area of irregular shapes?
4. How can you verify if a quadrilateral is a parallelogram or trapezoid using coordinates?
5. How does the area formula change for shapes with curved sides like circles or ellipses?

Tip: Always ensure that vertices are listed in a consecutive order (clockwise or counterclockwise) when using the shoelace formula.