To find the point $D(x, y)$ such that points $A(-3, 1)$, $B(4, 0)$, $C(0, -3)$, and $D$ form the corners of a square, we can follow these steps:

Step 1: Understanding the structure of a square

In a square:

• The diagonals are equal in length.
• The diagonals bisect each other at right angles.
• The opposite sides are equal and perpendicular.

Step 2: Calculate the midpoint of the diagonal

Let's assume that the points $A$ and $C$ are endpoints of one diagonal, and points $B$ and $D$ are endpoints of the other diagonal. The diagonals of a square bisect each other, meaning they share the same midpoint.

The midpoint of diagonal $AC$ is calculated as: $M_{AC} = \left( \frac{-3 + 0}{2}, \frac{1 + (-3)}{2} \right) = \left( \frac{-3}{2}, \frac{-2}{2} \right) = \left( -\frac{3}{2}, -1 \right)$

Similarly, the midpoint of diagonal $BD$ must also be $M_{BD} = \left( -\frac{3}{2}, -1 \right)$.

Step 3: Find the length of the diagonal

The length of the diagonal $AC$ is calculated using the distance formula: $d_{AC} = \sqrt{(0 - (-3))^2 + (-3 - 1)^2} = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Since $AC$ and $BD$ are diagonals of the square, $BD$ must also have the same length of 5 units.

Step 4: Use the midpoint and diagonal length to find point $D$

We now know:

• The midpoint $M_{BD}$ is $\left( -\frac{3}{2}, -1 \right)$.
• The distance from $B(4, 0)$ to $D$ must be 5 units, since the diagonals are equal in length.

Let's denote the coordinates of $D$ as $(x, y)$. Using the fact that $M_{BD}$ is the midpoint of $B(4, 0)$ and $D(x, y)$, we can set up the following equations: $\frac{4 + x}{2} = -\frac{3}{2}, \quad \frac{0 + y}{2} = -1$

Step 5: Solve for $x$ and $y$

Solving the first equation: $\frac{4 + x}{2} = -\frac{3}{2}$ Multiply both sides by 2: $4 + x = -3$ Solve for $x$: $x = -3 - 4 = -7$

Solving the second equation: $\frac{0 + y}{2} = -1$ Multiply both sides by 2: $y = -2$

Step 6: Conclusion

Thus, the coordinates of point $D$ are $(-7, -2)$.

Justification

• The diagonals $AC$ and $BD$ are equal in length (5 units).
• The midpoint of $AC$ and $BD$ is the same.
• This confirms that $A(-3, 1)$, $B(4, 0)$, $C(0, -3)$, and $D(-7, -2)$ are the vertices of a square.

Would you like more details or have any questions?

Here are 5 related questions:

1. How would the solution change if different points were used for the diagonal?
2. How can we verify if these points form a square without using diagonals?
3. What is the significance of the midpoint in geometry problems?
4. How would this problem be solved in three-dimensional space?
5. How would the distance formula be derived from the Pythagorean theorem?

Tip: Always check for perpendicularity of sides when confirming the shape is a square.