The co ordinates of the base BC of an isosceles triangle are `B(1,3)` and `C(-2,7)` then the co-ordinates of its vertex A are

Options

A. `(5/6,6)`
B. `(-7,1/8)`
C. `(1,6)`
D. `(-1/2,5)`

Answer

To find the coordinates of the vertex A of the isosceles triangle with base BC, where B(1, 3) and C(-2, 7), we can follow these steps:

### Step 1: Assign Coordinates
Let the coordinates of vertex A be (x, y). We already have the coordinates of points B and C:
- B(1, 3)
- C(-2, 7)

### Step 2: Use the Distance Formula
In an isosceles triangle, the lengths of the two sides from the vertex to the base are equal. Therefore, we can set up the equation:
\[ AB = AC \]

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

### Step 3: Set the Distances Equal
Since AB = AC, we can square both sides to eliminate the square roots:
\[ (x - 1)^2 + (y - 3)^2 = (x + 2)^2 + (y - 7)^2 \]

### Step 4: Expand Both Sides
Expanding both sides gives:
\[ (x^2 - 2x + 1) + (y^2 - 6y + 9) = (x^2 + 4x + 4) + (y^2 - 14y + 49) \]

### Step 5: Simplify the Equation
Combining like terms results in:
\[ x^2 - 2x + 1 + y^2 - 6y + 9 = x^2 + 4x + 4 + y^2 - 14y + 49 \]

This simplifies to:
\[ -2x - 6y + 10 = 4x - 14y + 53 \]

### Step 6: Rearranging the Equation
Rearranging gives:
\[ -2x - 6y + 10 - 4x + 14y - 53 = 0 \]
\[ -6x + 8y - 43 = 0 \]
or
\[ 6x - 8y + 43 = 0 \]

### Step 7: Solve for y in terms of x
We can express y in terms of x:
\[ 8y = 6x + 43 \]
\[ y = \frac{6}{8}x + \frac{43}{8} \]
\[ y = \frac{3}{4}x + \frac{43}{8} \]

### Step 8: Identify Possible Coordinates
Now we can find specific points (x, y) that satisfy this equation. We can check various values of x to find corresponding y values.

### Step 9: Check Possible Points
We can check the options given:
1. (5/6, 6)
2. (-7, 1/8)
3. (1, 6)
4. (-1/2, 5)

Substituting these points into the equation \( 6x - 8y + 43 = 0 \) will confirm if they satisfy the equation.

### Conclusion
All the points provided in the options satisfy the equation derived from the isosceles triangle condition.