To calculate the area of the rectangle shown on the coordinate plane, we need to first find the lengths of its sides. This rectangle is tilted, so we'll use the distance formula to find the lengths of the two diagonally positioned sides (the height and the width).

Step 1: Identify the coordinates of the four vertices of the rectangle:

The four corners (vertices) of the rectangle are approximately:

• $A(2, 3)$
• $B(6, 6)$
• $C(10, -5)$
• $D(6, -8)$

Step 2: Use the distance formula:

The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Step 3: Find the lengths of two sides (one horizontal, one vertical):

1. Length of side AB (between $A(2, 3)$ and $B(6, 6)$): $d_{AB} = \sqrt{(6 - 2)^2 + (6 - 3)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$
2. Length of side AD (between $A(2, 3)$ and $D(6, -8)$): $d_{AD} = \sqrt{(6 - 2)^2 + (-8 - 3)^2} = \sqrt{4^2 + (-11)^2} = \sqrt{16 + 121} = \sqrt{137}$

Step 4: Calculate the area:

The area of a rectangle is given by: $\text{Area} = \text{Length} \times \text{Width}$ Using the lengths we found: [ \text{Area} = 5 \times \sqrt{137}