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

Answer

To find the coordinates of vertex A of the isosceles triangle ABC, we can follow these steps:

### Step 1: Understand the Problem
We have an isosceles triangle ABC where the coordinates of points B and C are given as B(1, 3) and C(-2, 7). We need to find the coordinates of vertex A such that the lengths of sides AB and AC are equal.

### Step 2: Assign Coordinates to Vertex A
Let the coordinates of vertex A be (x, y).

### Step 3: Use the Distance Formula
We will use the distance formula to express the lengths of sides AB and AC:
- The distance AB can be calculated as:
\[ AB = \sqrt{(x - 1)^2 + (y - 3)^2} \]
- The distance AC can be calculated as:
\[ AC = \sqrt{(x + 2)^2 + (y - 7)^2} \]

### Step 4: Set the Distances Equal
Since AB = AC (because triangle ABC is isosceles), we can set the two distance formulas equal to each other:
\[ \sqrt{(x - 1)^2 + (y - 3)^2} = \sqrt{(x + 2)^2 + (y - 7)^2} \]

### Step 5: Square Both Sides
To eliminate the square roots, we square both sides:
\[ (x - 1)^2 + (y - 3)^2 = (x + 2)^2 + (y - 7)^2 \]

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

### Step 7: Simplify the Equation
Cancel \(x^2\) and \(y^2\) from both sides:
\[ -2x + 10 - 6y = 4x + 53 - 14y \]
Rearranging gives:
\[ -2x - 4x + 14y - 6y + 10 - 53 = 0 \]
This simplifies to:
\[ -6x + 8y - 43 = 0 \]

### Step 8: Rearranging the Equation
Rearranging gives us:
\[ 6x - 8y + 43 = 0 \]

### Step 9: Finding Possible Coordinates
To find specific coordinates for A, we can express y in terms of x:
\[ 8y = 6x + 43 \implies y = \frac{6x + 43}{8} \]

### Step 10: Choosing Values for x
We can choose different values for x to find corresponding y values. For example, if we let \(x = 5\):
\[ y = \frac{6(5) + 43}{8} = \frac{30 + 43}{8} = \frac{73}{8} = 9.125 \]

### Conclusion
Thus, one possible coordinate for vertex A is (5, 9.125). However, we can choose other values for x to find other coordinates for A.