A rectangle is drawn on a coordinate plane with vertices (0,0), (5,3), (7,0), and (3,-3). Using the coordinate, what is the exact are of the rectangle?

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!

A rectangle is drawn on a coordinate plane with vertices (0,0), (5,3), (7,0), and (3,-3). Using the coordinates, what is the exact area of the rectangle?

Learn how to calculate the exact area of a rectangle on a coordinate plane using the distance formula and perpendicular slopes. This problem covers key concepts in coordinate geometry and is suitable for students in Grades 8-10.

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