If the mid-point of the line segment joining the points A(3, 4), B(k, 6) is P(x, y) and x + y - 10 = 0, find the value of k.

Answer

To find the value of \( k \) given the points \( A(3, 4) \) and \( B(k, 6) \), and the condition that the midpoint \( P(x, y) \) lies on the line \( x + y - 10 = 0 \), we can follow these steps:

### Step 1: Find the Midpoint Coordinates
The formula for the midpoint \( P \) of a line segment joining two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:
\[ P\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
For points \( A(3, 4) \) and \( B(k, 6) \):
- \( x_1 = 3 \), \( y_1 = 4 \)
- \( x_2 = k \), \( y_2 = 6 \)

Thus, the coordinates of the midpoint \( P \) are:
\[ P\left(\frac{3 + k}{2}, \frac{4 + 6}{2}\right) \]
Calculating \( y \):
\[ y = \frac{4 + 6}{2} = \frac{10}{2} = 5 \]
So, the coordinates of \( P \) are:
\[ P\left(\frac{3 + k}{2}, 5\right) \]

### Step 2: Use the Condition of the Line
The midpoint \( P \) lies on the line defined by the equation:
\[ x + y - 10 = 0 \]
Substituting the coordinates of \( P \) into the equation:
\[ \frac{3 + k}{2} + 5 - 10 = 0 \]

### Step 3: Simplify the Equation
Now, simplify the equation:
\[ \frac{3 + k}{2} + 5 - 10 = 0 \]
This simplifies to:
\[ \frac{3 + k}{2} - 5 = 0 \]
\[ \frac{3 + k}{2} = 5 \]

### Step 4: Solve for \( k \)
Now, multiply both sides by 2 to eliminate the fraction:
\[ 3 + k = 10 \]
Subtract 3 from both sides:
\[ k = 10 - 3 \]
\[ k = 7 \]

### Conclusion
The value of \( k \) is \( 7 \).

---