To find the area of a quadrilateral given its vertices, we can use the following formula:

$\text{Area} = \frac{1}{2} \left| x_1 y_2 + x_2 y_3 + x_3 y_4 + x_4 y_1 - (y_1 x_2 + y_2 x_3 + y_3 x_4 + y_4 x_1) \right|$

Let’s assign the coordinates of the vertices as follows:

• $(x_1, y_1) = (1, 3)$
• $(x_2, y_2) = (-1, -6)$
• $(x_3, y_3) = (4, -3)$
• $(x_4, y_4) = (6, 1)$

Now, applying the coordinates to the formula:

$\text{Area} = \frac{1}{2} \left| (1)(-6) + (-1)(-3) + (4)(1) + (6)(3) - \left[ (3)(-1) + (-6)(4) + (-3)(6) + (1)(1) \right] \right|$

We will now calculate this step by step.The area of the quadrilateral with vertices at $(1, 3)$, $(-1, -6)$, $(4, -3)$, and $(6, 1)$ is $31.5$ square units.

Would you like any further explanation or details on this method?

Here are 5 related questions for further practice:

1. How would the area change if one vertex was shifted?
2. How do you find the area of a triangle given its vertices?
3. Can the same formula be used for concave quadrilaterals?
4. What is the difference between the formula for quadrilateral area and that for polygonal area?
5. How would you compute the perimeter of the same quadrilateral?

Tip: When working with polygons, always check the order of vertices (clockwise or counterclockwise) to ensure correct application of the formula.