To solve for the perimeter and the area of quadrilateral $ABCD$, we can use the coordinates provided and apply mathematical formulas.

1. Coordinates of the vertices:

The coordinates of the quadrilateral $ABCD$ are:

• $A(-5, 4)$
• $B(0, 3)$
• $C(4, -1)$
• $D(4, -5)$

2. Perimeter of the quadrilateral

The perimeter is the sum of the lengths of the sides of the quadrilateral. We will use the distance formula to calculate the lengths of the sides.

The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Side $AB$:

$AB = \sqrt{(0 - (-5))^2 + (3 - 4)^2} = \sqrt{(5)^2 + (-1)^2} = \sqrt{25 + 1} = \sqrt{26} \approx 5.10$

Side $BC$:

$BC = \sqrt{(4 - 0)^2 + (-1 - 3)^2} = \sqrt{(4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} \approx 5.66$

Side $CD$:

$CD = \sqrt{(4 - 4)^2 + (-5 - (-1))^2} = \sqrt{(0)^2 + (-4)^2} = \sqrt{16} = 4$

Side $DA$:

$DA = \sqrt{(4 - (-5))^2 + (-5 - 4)^2} = \sqrt{(9)^2 + (-9)^2} = \sqrt{81 + 81} = \sqrt{162} \approx 12.73$

Now, we can sum the lengths of the sides to find the perimeter:

$\text{Perimeter} = AB + BC + CD + DA = 5.10 + 5.66 + 4 + 12.73 = 27.49 \text{ units}$

3. Area of the quadrilateral

The area of a polygon with vertices $(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)$ can be calculated using the shoelace formula:

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

Using the coordinates of $A(-5, 4), B(0, 3), C(4, -1), D(4, -5)$:

$\text{Area} = \frac{1}{2} \left| (-5)(3) + (0)(-1) + (4)(-5) + (4)(4) - \left(4(0) + 3(4) + (-1)(4) + (-5)(-5)\right) \right|$ $= \frac{1}{2} \left| -15 + 0 - 20 + 16 - \left(0 + 12 - 4 + 25\right) \right|$ $= \frac{1}{2} \left| -19 - 33 \right| = \frac{1}{2} \times 52 = 26 \text{ square units}$

Final Answers:

• The perimeter of quadrilateral $ABCD$ is approximately 27.49 units.
• The area of quadrilateral $ABCD$ is 26 square units.

Would you like more details or have any questions? Here are some related questions:

1. How do we generalize the distance formula in 3D coordinates?
2. What are other methods to calculate the area of irregular polygons?
3. How do we derive the shoelace formula?
4. Can we use vector cross products to find the area of polygons?
5. What happens to the perimeter if we change one vertex position?

Tip: Always double-check your calculations by revisiting each step, especially in complex geometrical problems.