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

Options

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

Answer

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

### Step 1: Define the coordinates of vertex A
Let the coordinates of vertex A be (x, y).

### Step 2: Use the distance formula
Since triangle ABC is isosceles, the distances AB and AC must be equal. We can use the distance formula to express this mathematically.

The distance AB is given by:
\[ AB = \sqrt{(x - 1)^2 + (y - 3)^2} \]

The distance AC is given by:
\[ AC = \sqrt{(x + 2)^2 + (y - 7)^2} \]

### Step 3: Set the distances equal
Since AB = AC, we can set the two distance equations equal to each other:
\[ \sqrt{(x - 1)^2 + (y - 3)^2} = \sqrt{(x + 2)^2 + (y - 7)^2} \]

### Step 4: Square both sides to eliminate the square roots
Squaring both sides gives:
\[ (x - 1)^2 + (y - 3)^2 = (x + 2)^2 + (y - 7)^2 \]

### Step 5: Expand both sides
Expanding both sides:
Left side:
\[ (x^2 - 2x + 1) + (y^2 - 6y + 9) = x^2 + y^2 - 2x - 6y + 10 \]

Right side:
\[ (x^2 + 4x + 4) + (y^2 - 14y + 49) = x^2 + y^2 + 4x - 14y + 53 \]

### Step 6: Set the expanded equations equal
Now, we have:
\[ x^2 + y^2 - 2x - 6y + 10 = x^2 + y^2 + 4x - 14y + 53 \]

### Step 7: Simplify the equation
Cancel \(x^2\) and \(y^2\) from both sides:
\[ -2x - 6y + 10 = 4x - 14y + 53 \]

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

### Step 8: Rearranging the equation
This simplifies to:
\[ 6x - 8y + 43 = 0 \]

### Step 9: Finding possible coordinates for A
Now we can find possible coordinates (x, y) that satisfy this equation.

### Step 10: Check possible coordinates
For example, we can check the options (5/6, 6) and (-7, 1/8) to see if they satisfy the equation:
1. For (5/6, 6):
\[ 6(5/6) - 8(6) + 43 = 5 - 48 + 43 = 0 \] (satisfies)

2. For (-7, 1/8):
\[ 6(-7) - 8(1/8) + 43 = -42 - 1 + 43 = 0 \] (satisfies)

### Conclusion
The coordinates of vertex A can be (5/6, 6) or (-7, 1/8).