https://file.aiquickdraw.com/mathbot/file/1727437819994r61z7lkf.jpg

The image shows a math problem in German. Here's the translated version of the problem:

---

**Draw the points**  
- A (2|1),  
- B (2|3),  
- and C (-4|3) in a coordinate system.

**Tasks:**
1. **(a)** Add a point D so that the quadrilateral ABCD is a rectangle. Provide the coordinates of point D.
2. **(b)** Draw the diagonals AC and BD. Provide the coordinates of the intersection point P of the diagonals.
3. **(c)** Reflect point A over the y-axis. Provide the coordinates of the reflected point.

---

Let me walk you through solving the problem:

### Part (a):  
We need to determine the coordinates of point D to make ABCD a rectangle. Points A, B, and C are given as:

- \( A(2, 1) \)
- \( B(2, 3) \)
- \( C(-4, 3) \)

To form a rectangle, the opposite sides must be parallel and of equal length. Therefore, point D must be at \((-4, 1)\), as it ensures that \(AB \parallel CD\) and \(AD \parallel BC\), creating a rectangle.

So, the coordinates of point D are:  
**D (-4, 1)**

---

### Part (b):  
The diagonals of a rectangle intersect at their midpoints. The diagonals here are \(AC\) and \(BD\).

- Diagonal \(AC\) connects \(A(2,1)\) and \(C(-4,3)\).
- Diagonal \(BD\) connects \(B(2,3)\) and \(D(-4,1)\).

To find the intersection point, we calculate the midpoint of either diagonal:

- Midpoint formula:  
\[
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
- Using points \(A(2, 1)\) and \(C(-4, 3)\):
\[
P = \left( \frac{2 + (-4)}{2}, \frac{1 + 3}{2} \right) = (-1, 2)
\]

So, the intersection point \(P\) of the diagonals is:  
**P (-1, 2)**

---

### Part (c):  
To reflect point A over the y-axis, we change the sign of the x-coordinate of point A:

- Point \(A(2, 1)\) becomes \(A'(-2, 1)\) after reflection.

Thus, the coordinates of the reflected point are:  
**A' (-2, 1)**

---

Would you like any additional details or clarifications on the solution? Here are five related questions for further practice:

1. How do you find the slope of a line between two points?
2. What is the formula for calculating the distance between two points?
3. How would you determine if a given quadrilateral is a square instead of a rectangle?
4. How can you reflect a point over the x-axis?
5. How do you find the equation of a line through two given points?

**Tip:** For any rectangle, the diagonals are always equal in length and bisect each other at their midpoint.

Draw the points A (2|1), B (2|3), and C (-4|3) in a coordinate system. Add a point D to form a rectangle ABCD. Draw diagonals AC and BD, and find the intersection point P. Reflect point A over the y-axis.

This page guides on plotting points in coordinate geometry to form a rectangle, calculating the intersection of diagonals using the midpoint formula, and reflecting a point over the y-axis. Ideal for students in grades 8-10 exploring basic geometric constructions and transformations.

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