VII Answer the following 32. Plot the points $$ A(2,2) B(-2,2)(-2,-2) $$ and $$ D(2,-2) $$ on graph paper. Show that $$ A B C $$ is a square.

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1. **Plotting the Points**  
   The given points are  
   $$
   A(2,2),\quad B(-2,2),\quad C(-2,-2),\quad D(2,-2)
   $$  
   When plotted on graph paper, these points lie at the vertices of a quadrilateral.

2. **Checking the Lengths of the Sides**  
   We use the distance formula  
   $$
   d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   $$
   to calculate the side lengths.

- **Side $$AB$$:**  
     $$
     AB = \sqrt{((-2) - 2)^2 + (2 - 2)^2} = \sqrt{(-4)^2 + 0^2} = \sqrt{16} = 4
     $$
   - **Side $$BC$$:**  
     $$
     BC = \sqrt{((-2) - (-2))^2 + ((-2) - 2)^2} = \sqrt{0^2 + (-4)^2} = \sqrt{16} = 4
     $$
   - **Side $$CD$$:**  
     $$
     CD = \sqrt{(2 - (-2))^2 + ((-2) - (-2))^2} = \sqrt{(4)^2 + 0^2} = \sqrt{16} = 4
     $$
   - **Side $$DA$$:**  
     $$
     DA = \sqrt{(2 - 2)^2 + (2 - (-2))^2} = \sqrt{0^2 + (4)^2} = \sqrt{16} = 4
     $$

Since $$AB = BC = CD = DA = 4$$, all sides are equal.

3. **Verifying the Right Angles**  
   A square must have all interior angles equal to $$90^\circ$$. There are several ways to check this; one method is to examine the slopes of the adjacent sides.

- **Slope of $$AB$$:**  
     $$
     m_{AB} = \frac{2 - 2}{-2 - 2} = \frac{0}{-4} = 0
     $$
   - **Slope of $$BC$$:**  
     $$
     m_{BC} = \frac{-2 - 2}{-2 - (-2)} = \frac{-4}{0}
     $$
     The slope of $$BC$$ is undefined, indicating a vertical line.

Since the slope of $$AB$$ is $$0$$ (horizontal) and the slope of $$BC$$ is undefined (vertical), they are perpendicular. A similar check for the other pairs of consecutive sides will show that all adjacent sides are perpendicular.

4. **Conclusion**  
   Since the quadrilateral formed by $$A$$, $$B$$, $$C$$, and $$D$$ has four equal sides and every adjacent pair of sides is perpendicular, the quadrilateral is a square.
All sides of the quadrilateral $$ABCD$$ are equal in length, and each adjacent pair of sides is perpendicular. Therefore, $$ABCD$$ is a square.

VII Answer the following 32. Plot the points \( A(2,2) B(-2,2)(-2,-2) \) and \( D(2,-2) \) on graph paper. Show that \( A B C \) is a square.

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