To find the exact area of the rectangle given the coordinates of its vertices $(0,0)$, $(5,3)$, $(7,0)$, and $(3,-3)$, we need to determine the lengths of two adjacent sides of the rectangle.

Step 1: Identify Adjacent Vertices

From the vertices provided, let's check which pairs are adjacent:

• $(0,0)$ to $(5,3)$
• $(0,0)$ to $(7,0)$
• $(7,0)$ to $(3,-3)$
• $(5,3)$ to $(3,-3)$

Step 2: Calculate the Lengths of Two Adjacent Sides

Let's calculate the length of each side using the distance formula:

$\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

1. Side 1: Between $(0,0)$ and $(7,0)$

   $\text{Length} = \sqrt{(7 - 0)^2 + (0 - 0)^2} = \sqrt{49} = 7$

2. Side 2: Between $(0,0)$ and $(3,-3)$

   $\text{Length} = \sqrt{(3 - 0)^2 + (-3 - 0)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}$

Step 3: Verify Perpendicularity of Sides

To ensure that the identified lengths form the rectangle, check if the sides are perpendicular:

Slope of $(0,0)$ to $(7,0)$:

$\text{Slope} = \frac{0 - 0}{7 - 0} = 0$

Slope of $(0,0)$ to $(3, -3)$:

$\text{Slope} = \frac{-3 - 0}{3 - 0} = -1$

Since the slopes of these sides multiply to $-1$, they are perpendicular.

Step 4: Calculate the Area

Now, calculate the area using the formula for the area of a rectangle:

$\text{Area} = \text{Side 1} \times \text{Side 2} = 7 \times 3\sqrt{2} = 21\sqrt{2}$

So, the exact area of the rectangle is:

$\boxed{21\sqrt{2}}$

Would you like further details or have any questions about the steps?

Here are some related questions:

1. How do you find the area of a polygon using coordinates?
2. How can you verify that a quadrilateral is a rectangle using coordinates?
3. What are some other ways to find the area of irregular shapes on a coordinate plane?
4. How can you determine if four points form a parallelogram?
5. What are the properties of a rectangle in a coordinate plane?

Tip: Always verify that sides are perpendicular when checking for rectangles using coordinate geometry!