To find the area of the rectangle shown in the graph, we first need to identify the coordinates of its vertices and then use geometric methods.

The vertices seem to be at the following points (based on approximate grid markings):

1. (-1, 7)
2. (-5, 4)
3. (-2, -3)
4. (2, 0)

Steps to find the area:

1. Calculate the length of the sides: Since this is a rectangle but rotated, we need to find the lengths of the diagonally opposite sides using the distance formula:

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

   • For one side between points (-1, 7) and (-5, 4): $d_1 = \sqrt{((-5) - (-1))^2 + (4 - 7)^2} = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

   • For the other side between points (-5, 4) and (-2, -3): $d_2 = \sqrt{((-2) - (-5))^2 + ((-3) - 4)^2} = \sqrt{(3)^2 + (-7)^2} = \sqrt{9 + 49} = \sqrt{58} \approx 7.6$

2. Area of the rectangle: The area $A$ of a rectangle is given by the product of the lengths of two adjacent sides: $A = d_1 \times d_2 = 5 \times 7.6 = 38 \, \text{square units}$ Approximating the nearest given answer, the area is closest to 40 square units.

Would you like a detailed explanation or have any other questions?

Follow-up questions:

1. How do we calculate the distance between two points in geometry?
2. What is the formula for the area of a rotated rectangle?
3. What are some examples of applying the distance formula in real-world problems?
4. How does rotation affect the area of a rectangle?
5. Can the rectangle's diagonal be used directly to find the area?

Tip:

Always double-check the coordinates carefully when reading from a graph to ensure accurate distance calculations.