Understanding the Problem
To find the value of \( p \) such that point \( A(0, 2) \) is equidistant from points \( B(3, p) \) and \( C(p, 5) \), we will use the distance formula. The distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Calculating Distances
1. Distance from A to B:
The distance \( d_{AB} \) can be expressed as:
\[
d_{AB} = \sqrt{(3 - 0)^2 + (p - 2)^2} = \sqrt{9 + (p - 2)^2}
\]
2. Distance from A to C:
The distance \( d_{AC} \) is given by:
\[
d_{AC} = \sqrt{(p - 0)^2 + (5 - 2)^2} = \sqrt{p^2 + 9}
\]
Setting Distances Equal
Since \( A \) is equidistant from \( B \) and \( C \):
\[
d_{AB} = d_{AC}
\]
This leads to the equation:
\[
\sqrt{9 + (p - 2)^2} = \sqrt{p^2 + 9}
\]
Squaring Both Sides
Squaring both sides eliminates the square roots:
\[
9 + (p - 2)^2 = p^2 + 9
\]
Simplifying the Equation
Subtract \( 9 \) from both sides:
\[
(p - 2)^2 = p^2
\]
Expanding the left side:
\[
p^2 - 4p + 4 = p^2
\]
Subtract \( p^2 \) from both sides:
\[
-4p + 4 = 0
\]
Solving for p
Rearranging gives:
\[
4p = 4 \implies p = 1
\]
Conclusion
Thus, the value of \( p \) is:
\[
\boxed{1}
\]