If `A (-3, 2)` , `B(x, y)` and `C(1, 4)` are the vertices of an isosceles triangle with `AB=BC` . Find the value of `(2x+y)` .

Answer

To solve the problem, we need to find the value of \(2x + y\) given that the triangle formed by the points \(A(-3, 2)\), \(B(x, y)\), and \(C(1, 4)\) is isosceles with \(AB = BC\).

### Step 1: Write the distance formulas for \(AB\) and \(BC\)

The distance \(AB\) can be calculated using the distance formula:
\[ AB = \sqrt{(x - (-3))^2 + (y - 2)^2} = \sqrt{(x + 3)^2 + (y - 2)^2} \]

The distance \(BC\) is:
\[ BC = \sqrt{(x - 1)^2 + (y - 4)^2} \]

### Step 2: Set the distances equal to each other

Since \(AB = BC\), we can set the two distance equations equal:
\[ \sqrt{(x + 3)^2 + (y - 2)^2} = \sqrt{(x - 1)^2 + (y - 4)^2} \]

### Step 3: Square both sides to eliminate the square roots

Squaring both sides gives:
\[ (x + 3)^2 + (y - 2)^2 = (x - 1)^2 + (y - 4)^2 \]

### Step 4: Expand both sides

Expanding both sides:
\[ (x^2 + 6x + 9) + (y^2 - 4y + 4) = (x^2 - 2x + 1) + (y^2 - 8y + 16) \]

### Step 5: Simplify the equation

Combining like terms, we have:
\[ x^2 + 6x + 9 + y^2 - 4y + 4 = x^2 - 2x + 1 + y^2 - 8y + 16 \]

The \(x^2\) and \(y^2\) terms cancel out:
\[ 6x + 9 - 4y + 4 = -2x + 1 - 8y + 16 \]

This simplifies to:
\[ 6x + 13 - 4y = -2x + 17 - 8y \]

### Step 6: Rearranging the equation

Rearranging gives:
\[ 6x + 2x + 4y - 8y = 17 - 13 \]
\[ 8x - 4y = 4 \]

### Step 7: Divide the entire equation by 4

Dividing the equation by 4:
\[ 2x - y = 1 \]

### Step 8: Solve for \(y\)

From the equation \(2x - y = 1\), we can express \(y\) in terms of \(x\):
\[ y = 2x - 1 \]

### Step 9: Substitute \(y\) back into \(2x + y\)

Now, substituting \(y\) back into \(2x + y\):
\[ 2x + y = 2x + (2x - 1) = 4x - 1 \]

### Step 10: Conclusion

Thus, the value of \(2x + y\) is \(4x - 1\). However, we need a specific numerical value for \(x\) to find \(2x + y\).

### Final Calculation

Since we don't have a specific value for \(x\), we can express \(2x + y\) in terms of \(x\):
\[ 2x + y = 4x - 1 \]