If P (1, 2), Q (4, 6), R (5, 7) and S (a, b) are the vertices of a parallelogram PQRS, then

Options

A. a = 2, b = 4
B. a = 3, b = 4
C. a = 2, b = 3
D. a = 3, b = 5

Answer

To find the coordinates \( S(a, b) \) of the vertex of the parallelogram \( PQRS \) given the vertices \( P(1, 2) \), \( Q(4, 6) \), and \( R(5, 7) \), we can follow these steps:

### Step 1: Understand the properties of a parallelogram
In a parallelogram, the diagonals bisect each other. This means that the midpoint of diagonal \( PR \) will be the same as the midpoint of diagonal \( QS \).

### Step 2: Find the midpoint of diagonal \( PR \)
The coordinates of points \( P \) and \( R \) are \( P(1, 2) \) and \( R(5, 7) \). The formula for the midpoint \( M \) of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Applying this to points \( P \) and \( R \):
\[ M_{PR} = \left( \frac{1 + 5}{2}, \frac{2 + 7}{2} \right) = \left( \frac{6}{2}, \frac{9}{2} \right) = (3, 4.5) \]

### Step 3: Set up the equation for the midpoint of diagonal \( QS \)
Let the coordinates of point \( S \) be \( (a, b) \). The coordinates of point \( Q \) are \( Q(4, 6) \). The midpoint \( M_{QS} \) can be calculated as:
\[ M_{QS} = \left( \frac{4 + a}{2}, \frac{6 + b}{2} \right) \]

### Step 4: Equate the midpoints
Since the midpoints \( M_{PR} \) and \( M_{QS} \) are equal, we set them equal to each other:
\[ \left( \frac{4 + a}{2}, \frac{6 + b}{2} \right) = (3, 4.5) \]

This gives us two equations:
1. \( \frac{4 + a}{2} = 3 \)
2. \( \frac{6 + b}{2} = 4.5 \)

### Step 5: Solve for \( a \)
From the first equation:
\[ \frac{4 + a}{2} = 3 \]
Multiplying both sides by 2:
\[ 4 + a = 6 \]
Subtracting 4 from both sides:
\[ a = 2 \]

### Step 6: Solve for \( b \)
From the second equation:
\[ \frac{6 + b}{2} = 4.5 \]
Multiplying both sides by 2:
\[ 6 + b = 9 \]
Subtracting 6 from both sides:
\[ b = 3 \]

### Conclusion
The coordinates of point \( S \) are \( (a, b) = (2, 3) \).