To calculate the area of the rectangle, we can use the formula for the area of a rectangle, which is the product of its length and width. However, this rectangle is not aligned with the axes, so we need to use the distance formula to find the lengths of its sides.

We are given the coordinates of the vertices:

• $A(4, 2)$
• $B(10, 4)$
• $C(6, 16)$
• $D(0, 14)$

Step 1: Calculate the length of one side (AB)

Using the distance formula between points $A$ and $B$: $AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Substitute $A(4, 2)$ and $B(10, 4)$: $AB = \sqrt{(10 - 4)^2 + (4 - 2)^2} = \sqrt{6^2 + 2^2} = \sqrt{36 + 4} = \sqrt{40} \approx 6.32 \text{ units}$

Step 2: Calculate the length of the other side (AD)

Using the distance formula between points $A$ and $D$: $AD = \sqrt{(0 - 4)^2 + (14 - 2)^2} = \sqrt{(-4)^2 + (12)^2} = \sqrt{16 + 144} = \sqrt{160} \approx 12.65 \text{ units}$

Step 3: Calculate the area of the rectangle

Now that we know the lengths of two sides: $\text{Area} = AB \times AD = 6.32 \times 12.65 \approx 79.91$ Rounding to the nearest whole number: $\text{Area} \approx 80 \text{ square units}$

Thus, the approximate area of the rectangle is 80 square units.

Would you like more details on any part of the process?

Here are five questions that could extend this concept:

1. How would the process change if this were a parallelogram instead of a rectangle?
2. Can you calculate the area using a different pair of sides, such as $BC$ and $CD$?
3. How does the distance formula generalize to three-dimensional space?
4. What would happen to the area if the vertices were reflected across the x-axis?
5. How do we calculate the perimeter of the rectangle using the same coordinate points?

Tip: Always verify that the shape is a rectangle by confirming the opposite sides are parallel (using slopes) or by checking that the diagonals are equal.