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!