If the points A(a, -10), B(6, b), C (3, 16), D (2, -1) are the vertices of a parallelogram ABCD, find the values of a and b.

Answer

To find the values of \( a \) and \( b \) for the vertices of the parallelogram \( ABCD \) with points \( A(a, -10) \), \( B(6, b) \), \( C(3, 16) \), and \( D(2, -1) \), we will use the property that the diagonals of a parallelogram bisect each other. This means that the midpoints of the diagonals \( AC \) and \( BD \) must be the same.

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

The coordinates of points \( A \) and \( C \) are:
- \( A(a, -10) \)
- \( C(3, 16) \)

Using the midpoint formula, the midpoint \( M_{AC} \) of \( AC \) is given by:

\[ M_{AC} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{a + 3}{2}, \frac{-10 + 16}{2} \right) \]

Calculating the coordinates:

\[ M_{AC} = \left( \frac{a + 3}{2}, \frac{6}{2} \right) = \left( \frac{a + 3}{2}, 3 \right) \]

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

The coordinates of points \( B \) and \( D \) are:
- \( B(6, b) \)
- \( D(2, -1) \)

Using the midpoint formula, the midpoint \( M_{BD} \) of \( BD \) is given by:

\[ M_{BD} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{6 + 2}{2}, \frac{b - 1}{2} \right) \]

Calculating the coordinates:

\[ M_{BD} = \left( \frac{8}{2}, \frac{b - 1}{2} \right) = \left( 4, \frac{b - 1}{2} \right) \]

### Step 3: Set the midpoints equal to each other

Since the midpoints \( M_{AC} \) and \( M_{BD} \) are equal, we can set their coordinates equal:

1. For the x-coordinates:

\[ \frac{a + 3}{2} = 4 \]

Multiplying both sides by 2:

\[ a + 3 = 8 \]

Subtracting 3 from both sides:

\[ a = 5 \]

2. For the y-coordinates:

\[ 3 = \frac{b - 1}{2} \]

Multiplying both sides by 2:

\[ 6 = b - 1 \]

Adding 1 to both sides:

\[ b = 7 \]

### Conclusion

The values of \( a \) and \( b \) are:

\[ a = 5, \quad b = 7 \]