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 use the property of the diagonals of a parallelogram. The diagonals bisect each other, which means that the midpoints of the diagonals are the same.

### Step-by-Step Solution:

1. **Identify the Midpoints of the Diagonals:**
The midpoints of the diagonals \( PR \) and \( QS \) should be equal.
- The midpoint \( T \) of diagonal \( PR \) can be calculated using the midpoint formula:
\[ T = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
where \( P(1, 2) \) and \( R(5, 7) \).
\[ T_{PR} = \left( \frac{1 + 5}{2}, \frac{2 + 7}{2} \right) = \left( \frac{6}{2}, \frac{9}{2} \right) = (3, 4.5) \]

2. **Calculate the Midpoint of \( QS \):**
- The midpoint \( T \) of diagonal \( QS \) can also be calculated:
\[ T_{QS} = \left( \frac{4 + a}{2}, \frac{6 + b}{2} \right) \]

3. **Set the Midpoints Equal:**
Since \( T_{PR} = T_{QS} \), we can set the coordinates equal to each other:
\[ \left( \frac{4 + a}{2}, \frac{6 + b}{2} \right) = (3, 4.5) \]

4. **Create Equations from the Coordinates:**
From the x-coordinates:
\[ \frac{4 + a}{2} = 3 \]
Multiply both sides by 2:
\[ 4 + a = 6 \]
Subtract 4 from both sides:
\[ a = 2 \]

From the y-coordinates:
\[ \frac{6 + b}{2} = 4.5 \]
Multiply both sides by 2:
\[ 6 + b = 9 \]
Subtract 6 from both sides:
\[ b = 3 \]

5. **Conclusion:**
The coordinates of point \( S \) are \( S(2, 3) \).

### Final Answer:
Thus, the coordinates of point \( S \) are \( (2, 3) \).

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