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

Options

A. a =2, b = 4
B. a = 3,6 = 4
C. a =3,b = 3
D. a=30=5

Answer

To find the coordinates \( S(a, b) \) of the fourth vertex of the parallelogram \( PQRS \) given the vertices \( P(1, 2) \), \( Q(4, 6) \), and \( R(6, 7) \), we can use the property that the diagonals of a parallelogram bisect each other.

### Step 1: Find the midpoint of diagonal \( PR \)

The coordinates of points \( P \) and \( R \) are:
- \( P(1, 2) \)
- \( R(6, 7) \)

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 \( R \):
\[ M_{PR} = \left( \frac{1 + 6}{2}, \frac{2 + 7}{2} \right) = \left( \frac{7}{2}, \frac{9}{2} \right) \]

### Step 2: Find the midpoint of diagonal \( QS \)

The coordinates of point \( Q \) are:
- \( Q(4, 6) \)
- \( S(a, b) \)

Using the midpoint formula again for points \( Q \) and \( S \):
\[ M_{QS} = \left( \frac{4 + a}{2}, \frac{6 + b}{2} \right) \]

### Step 3: Set the midpoints equal

Since the midpoints of the diagonals bisect each other, we can set the midpoints equal to each other:
\[ \left( \frac{7}{2}, \frac{9}{2} \right) = \left( \frac{4 + a}{2}, \frac{6 + b}{2} \right) \]

This gives us two equations:
1. \( \frac{7}{2} = \frac{4 + a}{2} \)
2. \( \frac{9}{2} = \frac{6 + b}{2} \)

### Step 4: Solve for \( a \)

From the first equation:
\[ \frac{7}{2} = \frac{4 + a}{2} \]
Multiply both sides by 2:
\[ 7 = 4 + a \]
Subtracting 4 from both sides gives:
\[ a = 7 - 4 = 3 \]

### Step 5: Solve for \( b \)

From the second equation:
\[ \frac{9}{2} = \frac{6 + b}{2} \]
Multiply both sides by 2:
\[ 9 = 6 + b \]
Subtracting 6 from both sides gives:
\[ b = 9 - 6 = 3 \]

### Conclusion

Thus, the coordinates of point \( S \) are \( S(3, 3) \).

### Final Answer
The coordinates of the fourth vertex \( S \) are \( (3, 3) \).

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