In a parallelogram PQRS (taken in order), P is the point (-1, -1), Q is (8, 0) and R is (7, 5). Then S is the point :

Options

A. `(-1, 4)`
B. `(-2, 2)`
C. `(-2, (7)/(2))`
D. `(-2, 4)`

Answer

To find the coordinates of point S in the parallelogram PQRS, we can use the property that the diagonals of a parallelogram bisect each other.

### Step-by-Step Solution:

1. **Identify the Given Points**:
- P = (-1, -1)
- Q = (8, 0)
- R = (7, 5)

2. **Let the Coordinates of S be (x, y)**.

3. **Find the Midpoint of Diagonal PR**:
- The midpoint M of diagonal PR can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
- Here, \(P(-1, -1)\) and \(R(7, 5)\):
\[ M = \left( \frac{-1 + 7}{2}, \frac{-1 + 5}{2} \right) = \left( \frac{6}{2}, \frac{4}{2} \right) = (3, 2) \]

4. **Find the Midpoint of Diagonal QS**:
- The midpoint of diagonal QS should also be M:
\[ M = \left( \frac{8 + x}{2}, \frac{0 + y}{2} \right) \]
- Setting this equal to the midpoint we found:
\[ \left( \frac{8 + x}{2}, \frac{0 + y}{2} \right) = (3, 2) \]

5. **Set Up the Equations**:
- From the x-coordinates:
\[ \frac{8 + x}{2} = 3 \]
- From the y-coordinates:
\[ \frac{0 + y}{2} = 2 \]

6. **Solve for x**:
- Multiply both sides of the x-coordinate equation by 2:
\[ 8 + x = 6 \]
- Solve for x:
\[ x = 6 - 8 = -2 \]

7. **Solve for y**:
- Multiply both sides of the y-coordinate equation by 2:
\[ y = 4 \]

8. **Conclusion**:
- The coordinates of point S are:
\[ S = (-2, 4) \]

### Final Answer:
The coordinates of point S are (-2, 4).

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