Fill in the blanks:
If P(2, 4), Q(0, 3), R (3, 6) and S(a, b) are vertices of a parallelogram then the value of a + b is ..........

Answer

To find the value of \( a + b \) for the vertices of the parallelogram \( P(2, 4) \), \( Q(0, 3) \), \( R(3, 6) \), and \( S(a, b) \), we will use the property that the diagonals of a parallelogram bisect each other.

### Step 1: Identify the midpoints of the diagonals
Let the midpoint of diagonal \( PR \) be \( O \) and the midpoint of diagonal \( QS \) also be \( O \).

### Step 2: Calculate the midpoint of diagonal \( PR \)
The coordinates of points \( P \) and \( R \) are:
- \( P(2, 4) \)
- \( R(3, 6) \)

Using the midpoint formula:
\[ O = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
we find:
\[ O = \left( \frac{2 + 3}{2}, \frac{4 + 6}{2} \right) = \left( \frac{5}{2}, \frac{10}{2} \right) = \left( \frac{5}{2}, 5 \right) \]

### Step 3: Calculate the midpoint of diagonal \( QS \)
The coordinates of points \( Q \) and \( S \) are:
- \( Q(0, 3) \)
- \( S(a, b) \)

Using the midpoint formula again:
\[ O = \left( \frac{0 + a}{2}, \frac{3 + b}{2} \right) \]

### Step 4: Set the midpoints equal to each other
Since both midpoints are equal, we can set the x-coordinates and y-coordinates equal:
1. For the x-coordinates:
\[ \frac{0 + a}{2} = \frac{5}{2} \]
Multiplying both sides by 2:
\[ a = 5 \]

2. For the y-coordinates:
\[ \frac{3 + b}{2} = 5 \]
Multiplying both sides by 2:
\[ 3 + b = 10 \]
Solving for \( b \):
\[ b = 10 - 3 = 7 \]

### Step 5: Calculate \( a + b \)
Now that we have \( a = 5 \) and \( b = 7 \):
\[ a + b = 5 + 7 = 12 \]

Thus, the value of \( a + b \) is **12**.

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