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To find the area of rhombus ABCD, we can use the formula:

\[
\text{Area of a rhombus} = \frac{1}{2} \times d_1 \times d_2
\]

Where:
- \( d_1 \) is the length of one diagonal (AC)
- \( d_2 \) is the length of the other diagonal (BD)

### Step 1: Find the coordinates of the points
- \( A(4, 0) \)
- \( B(2, -2) \)
- \( C(1, 4) \)
- \( D(5, -2) \)

### Step 2: Calculate the length of diagonal AC
Use the distance formula for diagonal \( AC \):
\[
d_1 = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
For points \( A(4, 0) \) and \( C(1, 4) \):
\[
d_1 = \sqrt{(4 - 1)^2 + (0 - 4)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]

### Step 3: Calculate the length of diagonal BD
For points \( B(2, -2) \) and \( D(5, -2) \):
\[
d_2 = \sqrt{(5 - 2)^2 + (-2 - (-2))^2} = \sqrt{3^2 + 0^2} = \sqrt{9} = 3
\]

### Step 4: Calculate the area
Now, use the formula for the area of the rhombus:
\[
\text{Area} = \frac{1}{2} \times 5 \times 3 = \frac{15}{2} = 7.5 \, \text{units}^2
\]

Thus, the area of rhombus ABCD is **7.5 square units**.

Would you like further details or have any questions?

### Related Questions:
1. How do we determine the type of quadrilateral using just its coordinates?
2. How is the distance formula derived from the Pythagorean theorem?
3. Can we calculate the area of a rhombus if the side lengths are given instead of diagonals?
4. What are the conditions for a quadrilateral to be a rhombus?
5. How does the area formula change for other parallelograms?

### Tip:
Always check the perpendicularity of diagonals in a rhombus; they should bisect each other at 90 degrees!

What is the area of rhombus ABCD given the coordinates of the vertices A(4, 0), B(2, -2), C(1, 4), and D(5, -2)?

Learn how to calculate the area of rhombus ABCD using its diagonals. This problem uses coordinate geometry and the distance formula to find the lengths of the diagonals, leading to the calculation of the area. Perfect for students in Grades 9-11.

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