If the point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), find p.

Answer

To solve the problem, we need to find the value of \( p \) such that the point \( A(0, 2) \) is equidistant from the points \( B(3, p) \) and \( C(p, 5) \).

### Step-by-step Solution:

1. **Understanding the Problem**:
We need to find \( p \) such that the distance \( AB \) is equal to the distance \( AC \).

2. **Distance Formula**:
The distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

3. **Calculate Distance \( AB \)**:
- The coordinates of \( A \) are \( (0, 2) \) and the coordinates of \( B \) are \( (3, p) \).
- Using the distance formula:
\[ AB = \sqrt{(3 - 0)^2 + (p - 2)^2} = \sqrt{3^2 + (p - 2)^2} = \sqrt{9 + (p - 2)^2} \]

4. **Calculate Distance \( AC \)**:
- The coordinates of \( C \) are \( (p, 5) \).
- Using the distance formula:
\[ AC = \sqrt{(p - 0)^2 + (5 - 2)^2} = \sqrt{p^2 + 3^2} = \sqrt{p^2 + 9} \]

5. **Set the Distances Equal**:
Since \( A \) is equidistant from \( B \) and \( C \), we set the distances equal:
\[ \sqrt{9 + (p - 2)^2} = \sqrt{p^2 + 9} \]

6. **Square Both Sides**:
To eliminate the square roots, we square both sides:
\[ 9 + (p - 2)^2 = p^2 + 9 \]

7. **Simplify the Equation**:
- Expanding the left side:
\[ 9 + (p^2 - 4p + 4) = p^2 + 9 \]
- This simplifies to:
\[ p^2 - 4p + 13 = p^2 + 9 \]

8. **Eliminate \( p^2 \)**:
Subtract \( p^2 \) from both sides:
\[ -4p + 13 = 9 \]

9. **Solve for \( p \)**:
- Rearranging gives:
\[ -4p = 9 - 13 \]
\[ -4p = -4 \]
\[ p = 1 \]

10. **Conclusion**:
The value of \( p \) is \( 1 \).