If `P(4,7), Q(7,2) R(a,b)` and `S(3,8)` are the vertices of a parallelogram then find the value of a and b.

Answer

To find the values of \( a \) and \( b \) for the vertices of the parallelogram \( P(4,7), Q(7,2), R(a,b), S(3,8) \), we can use the property that the diagonals of a parallelogram bisect each other. This means that the midpoints of the diagonals will be the same.

### Step-by-Step Solution:

1. **Identify the Midpoint of Diagonal PQ:**
The coordinates of points \( P \) and \( Q \) are \( P(4, 7) \) and \( Q(7, 2) \).
The midpoint \( O \) of \( PQ \) can be calculated using the midpoint formula:
\[ O = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of \( P \) and \( Q \):
\[ O = \left( \frac{4 + 7}{2}, \frac{7 + 2}{2} \right) = \left( \frac{11}{2}, \frac{9}{2} \right) \]

2. **Identify the Midpoint of Diagonal RS:**
The coordinates of points \( R \) and \( S \) are \( R(a, b) \) and \( S(3, 8) \).
The midpoint \( O' \) of \( RS \) can be calculated similarly:
\[ O' = \left( \frac{a + 3}{2}, \frac{b + 8}{2} \right) \]

3. **Set the Midpoints Equal:**
Since the midpoints \( O \) and \( O' \) are equal, we can set the coordinates equal to each other:
\[ \frac{a + 3}{2} = \frac{11}{2} \quad \text{(1)} \]
\[ \frac{b + 8}{2} = \frac{9}{2} \quad \text{(2)} \]

4. **Solve for \( a \):**
From equation (1):
\[ a + 3 = 11 \quad \Rightarrow \quad a = 11 - 3 = 8 \]

5. **Solve for \( b \):**
From equation (2):
\[ b + 8 = 9 \quad \Rightarrow \quad b = 9 - 8 = 1 \]

6. **Final Values:**
Thus, the values of \( a \) and \( b \) are:
\[ a = 8, \quad b = 1 \]

### Final Answer:
The values of \( a \) and \( b \) are \( a = 8 \) and \( b = 1 \).

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