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

Answer

To solve the problem step by step, we will follow the process outlined in the video transcript.

### Step 1: Understand the Midpoint Formula
The midpoint \( M \) of a line segment joining two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
In our case, the coordinates of point \( A \) are \( (3, 4) \) and the coordinates of point \( B \) are \( (k, 6) \).

### Step 2: Calculate the Midpoint
Using the midpoint formula:
- The x-coordinate of the midpoint \( x \) is:
\[ x = \frac{3 + k}{2} \]
- The y-coordinate of the midpoint \( y \) is:
\[ y = \frac{4 + 6}{2} = \frac{10}{2} = 5 \]

### Step 3: Set Up the Equation
We are given that \( x + y = 10 \). Substituting the values we found:
\[ \frac{3 + k}{2} + 5 = 10 \]

### Step 4: Solve for \( k \)
Now, we will solve the equation:
1. Subtract 5 from both sides:
\[ \frac{3 + k}{2} = 10 - 5 \]
\[ \frac{3 + k}{2} = 5 \]
2. Multiply both sides by 2 to eliminate the fraction:
\[ 3 + k = 10 \]
3. Subtract 3 from both sides:
\[ k = 10 - 3 \]
\[ k = 7 \]

### Step 5: Find the Coordinates of Point B
Now that we have \( k \), we can find the coordinates of point \( B \):
\[ B = (k, 6) = (7, 6) \]

### Step 6: Calculate the Length of Segment AB
We will use the distance formula to find the length of segment \( AB \):
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of points \( A(3, 4) \) and \( B(7, 6) \):
\[ d = \sqrt{(7 - 3)^2 + (6 - 4)^2} \]
\[ d = \sqrt{(4)^2 + (2)^2} \]
\[ d = \sqrt{16 + 4} \]
\[ d = \sqrt{20} = 2\sqrt{5} \]

### Final Answer
The value of \( k \) is \( 7 \) and the length of the line segment \( AB \) is \( 2\sqrt{5} \).

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