`A B C` is an isosceles triangle. If the coordinates of the base are `B(1,3)` and `C(-2,7)` , the coordinates of vertex `A`

Options

A. (1,6)
B. `(-1//2,5) `
C. `(-5//6,6) `
D. none of these

Explanation

To find the coordinates of vertex A in the isosceles triangle ABC with base BC, we will follow these steps:

### Step 1: Identify the coordinates of points B and C
The coordinates of point B are \( B(1, 3) \) and the coordinates of point C are \( C(-2, 7) \).

### Step 2: Set up the coordinates for point A
Let the coordinates of point A be \( A(x, y) \).

### Step 3: Use the distance formula
Since triangle ABC is isosceles, we know that the lengths of sides AB and AC are equal. We will use the distance formula to express this condition:

\[ AB = AC \]

Using the distance formula, we can express the lengths as follows:

\[ AB = \sqrt{(x - 1)^2 + (y - 3)^2} \]
\[ AC = \sqrt{(x + 2)^2 + (y - 7)^2} \]

### Step 4: Set the lengths equal and square both sides
Setting the lengths equal gives us:

\[ \sqrt{(x - 1)^2 + (y - 3)^2} = \sqrt{(x + 2)^2 + (y - 7)^2} \]

Squaring both sides removes the square roots:

\[ (x - 1)^2 + (y - 3)^2 = (x + 2)^2 + (y - 7)^2 \]

### Step 5: Expand both sides
Expanding both sides results in:

\[ (x^2 - 2x + 1) + (y^2 - 6y + 9) = (x^2 + 4x + 4) + (y^2 - 14y + 49) \]

### Step 6: Simplify the equation
Cancelling \( x^2 \) and \( y^2 \) from both sides gives:

\[ -2x + 10 = 4x + 53 - 14y \]

Rearranging terms leads to:

\[ -2x - 4x + 14y + 10 - 53 = 0 \]

This simplifies to:

\[ -6x + 14y - 43 = 0 \]

### Step 7: Rearranging the equation
We can rearrange this to:

\[ 6x - 14y + 43 = 0 \]

### Step 8: Check the options
Now, we will check which of the given options satisfies this equation:

1. **Option 1: \( (1, 6) \)**
\[ 6(1) - 14(6) + 43 = 6 - 84 + 43 = -35 \quad (\text{not valid}) \]

2. **Option 2: \( \left(-\frac{1}{2}, 5\right) \)**
\[ 6\left(-\frac{1}{2}\right) - 14(5) + 43 = -3 - 70 + 43 = -30 \quad (\text{not valid}) \]

3. **Option 3: \( \left(-\frac{5}{6}, 6\right) \)**
\[ 6\left(-\frac{5}{6}\right) - 14(6) + 43 = -5 - 84 + 43 = -46 \quad (\text{not valid}) \]

4. **Option 4: None of these**
Since none of the options satisfy the equation, the answer is:

### Final Answer
The coordinates of vertex A do not match any of the options provided.

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