Question: Find a point D(x, y) such that the points A(−3, 1), B(4, 0), C(0, −3) and D are the corners of a square. Justify your answer. 

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.

Find a point D(x, y) such that the points A(−3, 1), B(4, 0), C(0, −3) and D are the corners of a square. Justify your answer.

This problem guides you through finding the fourth vertex of a square given three points. Learn how to use the midpoint and distance formulas to solve this problem step-by-step. Suitable for students in Grades 9-12.

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