The fourth vertex of a rectangle whose other vertices are (4, 1) (7, 4) and (13, -2) is

Options

A. (10, -5)
B. (10, 5)
C. (-10, 5)
D. (-10, -5)

Answer

To find the fourth vertex of a rectangle given three vertices, we can use the properties of a rectangle where opposite sides are equal and the diagonals bisect each other. The three given vertices are A(4, 1), B(7, 4), and C(13, -2). We will denote the fourth vertex as D(x, y).

### Step-by-Step Solution:

1. **Identify the Given Points**:
The vertices of the rectangle are A(4, 1), B(7, 4), and C(13, -2). We need to find the coordinates of the fourth vertex D(x, y).

2. **Use the Midpoint Formula**:
The diagonals of a rectangle bisect each other. Therefore, the midpoint of diagonal AC should equal the midpoint of diagonal BD.
- Midpoint of AC = \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)
- Midpoint of BD = \(\left(\frac{x_3 + x_4}{2}, \frac{y_3 + y_4}{2}\right)\)

Let's calculate the midpoint of AC:
- A(4, 1) and C(13, -2)
- Midpoint of AC = \(\left(\frac{4 + 13}{2}, \frac{1 + (-2)}{2}\right) = \left(\frac{17}{2}, \frac{-1}{2}\right)\)

3. **Set Up the Equation for Midpoint of BD**:
Let D be (x, y). The coordinates of B are (7, 4).
- Midpoint of BD = \(\left(\frac{7 + x}{2}, \frac{4 + y}{2}\right)\)

Setting the midpoints equal:
\[ \frac{7 + x}{2} = \frac{17}{2} \quad \text{and} \quad \frac{4 + y}{2} = \frac{-1}{2} \]

4. **Solve for x**:
From the first equation:
\[ 7 + x = 17 \implies x = 17 - 7 = 10 \]

5. **Solve for y**:
From the second equation:
\[ 4 + y = -1 \implies y = -1 - 4 = -5 \]

6. **Conclusion**:
The coordinates of the fourth vertex D are (10, -5).

### Final Answer:
The fourth vertex of the rectangle is D(10, -5).