Introduction
To determine the value of \( k \) such that the point \( (0,2) \) is equidistant from the points \( (3,k) \) and \( (k,5) \), we will use the distance formula. The distance formula between two points \( (x_1,y_1) \) and \( (x_2,y_2) \) is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Step 1: Calculate Distances
- Distance from \( (0,2) \) to \( (3,k) \):
\[
d_1 = \sqrt{(3 - 0)^2 + (k - 2)^2} = \sqrt{9 + (k - 2)^2}
\]
- Distance from \( (0,2) \) to \( (k,5) \):
\[
d_2 = \sqrt{(k - 0)^2 + (5 - 2)^2} = \sqrt{k^2 + 9}
\]
Step 2: Set Distances Equal
Since the point \( (0,2) \) is equidistant from both points, we can set the distances equal:
\[
\sqrt{9 + (k - 2)^2} = \sqrt{k^2 + 9}
\]
Step 3: Square Both Sides
Squaring both sides removes the square roots:
\[
9 + (k - 2)^2 = k^2 + 9
\]
Step 4: Simplify
Subtract \( 9 \) from both sides:
\[
(k - 2)^2 = k^2
\]
Expanding the left side:
\[
k^2 - 4k + 4 = k^2
\]
Step 5: Solve for \( k \)
Subtract \( k^2 \) from both sides:
\[
-4k + 4 = 0
\]
Solving for \( k \):
\[
4k = 4 \implies k = 1
\]
Conclusion
Thus, the value of \( k \) for which the point \( (0,2) \) is equidistant from the points \( (3,k) \) and \( (k,5) \) is:
\[
\boxed{1}
\]