To determine whether quadrilateral $ABCD$ is a square, we will use the distance formula to calculate the lengths of all four sides of the quadrilateral. For a quadrilateral to be a square, all four sides must be of equal length, and the diagonals must also be equal in length.

Step 1: Identify the coordinates of the vertices

From the image, the approximate coordinates of the vertices are:

• $A = (-2, 2)$
• $B = (1, 3)$
• $C = (2, 0)$
• $D = (-1, -1)$

Step 2: Apply the Distance Formula

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

1. Length of $AB$: $AB = \sqrt{(1 - (-2))^2 + (3 - 2)^2} = \sqrt{(3)^2 + (1)^2} = \sqrt{9 + 1} = \sqrt{10}$

2. Length of $BC$: $BC = \sqrt{(2 - 1)^2 + (0 - 3)^2} = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10}$

3. Length of $CD$: $CD = \sqrt{(2 - (-1))^2 + (0 - (-1))^2} = \sqrt{(3)^2 + (1)^2} = \sqrt{9 + 1} = \sqrt{10}$

4. Length of $DA$: $DA = \sqrt{(-1 - (-2))^2 + (-1 - 2)^2} = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10}$

Step 3: Check the Diagonals

For a square, the diagonals should also be equal in length.

1. Length of $AC$: $AC = \sqrt{(2 - (-2))^2 + (0 - 2)^2} = \sqrt{(4)^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20}$

2. Length of $BD$: $BD = \sqrt{(1 - (-1))^2 + (3 - (-1))^2} = \sqrt{(2)^2 + (4)^2} = \sqrt{4 + 16} = \sqrt{20}$

Conclusion:

• All four sides $AB$, $BC$, $CD$, and $DA$ have the same length $\sqrt{10}$.
• Both diagonals $AC$ and $BD$ have the same length $\sqrt{20}$.

Despite all sides and diagonals being equal, this quadrilateral cannot be a square because the lengths of the diagonals should be $\sqrt{2}$ times the side length for a true square. Here, the diagonals are longer than this ratio. Therefore, $ABCD$ is a rhombus but not a square.

Would you like further details or have any questions?

Related Questions:

1. What is the specific condition for a rhombus to be a square?
2. How do we calculate the angles of quadrilateral $ABCD$ to further confirm its shape?
3. How does the orientation of a quadrilateral on the coordinate plane affect its classification?
4. Can you determine the area of quadrilateral $ABCD$ using the coordinates?
5. What is the significance of the diagonals' lengths in classifying quadrilaterals?

Tip:

To distinguish between a rhombus and a square, remember that a square must have equal diagonals that bisect each other at right angles, whereas a rhombus only requires equal side lengths with diagonals that bisect at any angle.